Similar shapes

1. Similarity


ID is: 3071 Seed is: 1615

Identifying similar polygons

The diagram below shows two polygons. The figure is drawn to scale.

  1. Are the shapes here similar to one another (select from the choices below)?
  2. Select the correct reason which explains the first answer.
shape A shape B
Answer:
  1. Are the shapes similar?
  2. The reason is:
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Similar objects must have the same shape, but might not be the same size.


STEP: Question 1: Compare the angles and sides of the two shapes
[−1 point ⇒ 1 / 2 points left]

The question shows us two shapes, and we need to decide if they are similar or not. In maths, two shapes are similar if corresponding angles are equal and if corresponding sides are all proportional (have the same ratio). In other words, if two objects are similar to each other, one of them can be "zoomed in or out" to make it identical to the other one.

These shapes are similar: the angles are all 90 degrees, and the sides are proportional because the sides of each shape are all equal: 918=918. In fact, any two equilateral squares must be similar.

shape A shape B

These two shapes are similar. The correct choice from the list is: Yes.


STEP: Question 2: Choose the correct reason or reasons
[−1 point ⇒ 0 / 2 points left]

As noted in the above explanation, the shapes are similar. This can only be true if the corresponding angles are equal and the sides are all proportional: both must be true for similarity.

Therefore, the correct choice from the list is: Both of the above


Submit your answer as: and

ID is: 3071 Seed is: 7410

Identifying similar polygons

The diagram below shows two polygons. The figure is drawn to scale.

  1. Are the shapes here similar to one another (select from the choices below)?
  2. Select the correct reason which explains the first answer.
shape A shape B
Answer:
  1. Are the shapes similar?
  2. The reason is:
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Similar objects must have the same shape, but might not be the same size.


STEP: Question 1: Compare the angles and sides of the two shapes
[−1 point ⇒ 1 / 2 points left]

The question shows us two shapes, and we need to decide if they are similar or not. In maths, two shapes are similar if corresponding angles are equal and if corresponding sides are all proportional (have the same ratio). In other words, if two objects are similar to each other, one of them can be "zoomed in or out" to make it identical to the other one.

The shapes here, which are kites, are not similar: corresponding angles are equal, but the sides are not proportional: 12 is not equal to 33.

shape A shape B

These two shapes are not similar. The correct choice from the list is: No.


STEP: Question 2: Choose the correct reason or reasons
[−1 point ⇒ 0 / 2 points left]

Therefore, the correct choice from the list is: The sides are not all proportional


Submit your answer as: and

ID is: 3071 Seed is: 2630

Identifying similar polygons

The diagram below shows two polygons. The figure is drawn to scale.

  1. Determine whether the two polygons shown are similar or not.
  2. What is the ratio of the size of polygon A to polygon B. Give your answer as a fraction. If the polygons are not similar, write 'None' for the ratio.
polygon A polygon B
Answer:
  1. Are the shapes similar?
  2. The ratio of the lengths is:
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Similar objects must have the same shape, but might not be the same size.


STEP: Question 1: Compare the angles and sides of the two polygons
[−1 point ⇒ 1 / 2 points left]

The question shows us two shapes, and we need to decide if they are similar or not. In maths, two shapes are similar if corresponding angles are equal and if corresponding sides are all proportional (have the same ratio). In other words, if two objects are similar to each other, one of them can be "zoomed in or out" to make it identical to the other one.

These polygons, which are kites, are similar. We can see that the four angles are all equal, and the sides are all proportional: 13=39.

polygon A polygon B

These two polygons are similar. The correct choice from the list is: Yes.


STEP: Question 2: Calculate the ratio of the shapes (if they are similar)
[−1 point ⇒ 0 / 2 points left]

Similar shapes always have proportional sides. We can use any pair of corresponding sides to find the ratio: 13.

The correct answer is: 13.


Submit your answer as: and

ID is: 4283 Seed is: 9494

Prove similarity in adjacent triangles

Consider the following diagram:

Prove that the triangles are similar.

Adebankole has already answered the question, and his proof is written below. But, Adebankole has made a mistake. Look carefully at his proof and identify where he has made his mistake.

Line Proof
(a) In ΔCAD and ΔABD:
(b) 1. CAAB=4520=2.25
(c) 2. DBDA=188=2.25
(d) 3. CDAD=40.518=2.25
(e) ΔCAD|||ΔABD (sides of Δ in prop)
Answer:

The mistake is on Line .

Replace this line with

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Check each line of the proof carefully. Are the sides of the triangles being divided in a consistent order? Is the similarity stated correctly, with the correct reason?


STEP: Identify the error in the proof
[−1 point ⇒ 2 / 3 points left]

When we use sides of Δ in prop to prove that two triangles are similar, we must divide the sides in the same way each time.

Here, Adebankole divided the sides in ΔCAD by the sides in ΔABD on Lines (b) and (d). But, on Line (c), Adebankole wrote the sides the other way around. So, the mistake is on Line (c).

NOTE: Adebankole substituted in the values for DADB, which is why he got the correct answer for the proportionality constant. But, this substitution did not match the labels he had written.

STEP: Correct the proof
[−2 points ⇒ 0 / 3 points left]

Adebankole should have divided the sides of ΔCAD by the sides of ΔABD every time.

The correct proof is:

Line Proof
(a) In ΔCAD and ΔABD:
(b) 1. CAAB=4520=2.25
(c) 2. DADB=188=2.25
(d) 3. CDAD=40.518=2.25
(e) ΔCAD|||ΔABD (sides of Δ in prop)

Submit your answer as: and

ID is: 4283 Seed is: 1812

Prove similarity in adjacent triangles

Consider the following diagram:

Prove that the triangles are similar.

Chibuike has already answered the question, and his proof is written below. But, Chibuike has made a mistake. Look carefully at his proof and identify where he has made his mistake.

Line Proof
(a) In ΔPRS and ΔPQR:
(b) 1. SPRP=13.59=1.5
(c) 2. PRPQ=96=1.5
(d) 3. RQSR=7.55=1.5
(e) ΔPRS|||ΔPQR (sides of Δ in prop)
Answer:

The mistake is on Line .

Replace this line with

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Check each line of the proof carefully. Are the sides of the triangles being divided in a consistent order? Is the similarity stated correctly, with the correct reason?


STEP: Identify the error in the proof
[−1 point ⇒ 2 / 3 points left]

When we use sides of Δ in prop to prove that two triangles are similar, we must divide the sides in the same way each time.

Here, Chibuike divided the sides in ΔPRS by the sides in ΔPQR on Lines (b) and (c). But, on Line (d), Chibuike wrote the sides the other way around. So, the mistake is on Line (d).

NOTE: Chibuike substituted in the values for SRRQ, which is why he got the correct answer for the proportionality constant. But, this substitution did not match the labels he had written.

STEP: Correct the proof
[−2 points ⇒ 0 / 3 points left]

Chibuike should have divided the sides of ΔPRS by the sides of ΔPQR every time.

The correct proof is:

Line Proof
(a) In ΔPRS and ΔPQR:
(b) 1. SPRP=13.59=1.5
(c) 2. PRPQ=96=1.5
(d) 3. SRRQ=7.55=1.5
(e) ΔPRS|||ΔPQR (sides of Δ in prop)

Submit your answer as: and

ID is: 4283 Seed is: 6871

Prove similarity in adjacent triangles

Consider the following diagram:

Prove that the triangles are similar.

Chichi has already answered the question, and her proof is written below. But, Chichi has made a mistake. Look carefully at her proof and identify where she has made her mistake.

Line Proof
(a) In ΔLJK and ΔMLK:
(b) 1. JKLK=87.535=2.5
(c) 2. KLKM=3514=2.5
(d) 3. JLLM=9036=2.5
(e) ΔLJKΔMLK (sides of Δ in prop)
Answer:

The mistake is on Line .

Replace this line with

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Check each line of the proof carefully. Are the sides of the triangles being divided in a consistent order? Is the similarity stated correctly, with the correct reason?


STEP: Identify the error in the proof
[−1 point ⇒ 2 / 3 points left]

Chichi has proved that the triangles are similar because their sides are in proportion. But, on Line (e), Chichi has used the symbol for congruency: . So, the mistake is on Line (e).

TIP: The symbol for congruency () looks like the equals sign (=) but with an extra line. We can think of congruent triangles as being equal in every way. So, you can remember the symbol for congruency because it is almost like the equals sign.

STEP: Correct the proof
[−2 points ⇒ 0 / 3 points left]

Chichi should have used the symbol for similar triangles: |||.

The correct proof is:

Line Proof
(a) In ΔLJK and ΔMLK:
(b) 1. JKLK=87.535=2.5
(c) 2. KLKM=3514=2.5
(d) 3. JLLM=9036=2.5
(e) ΔLJK|||ΔMLK (sides of Δ in prop)

Submit your answer as: and

ID is: 4289 Seed is: 6590

Consequences of similarity

In the diagram below, ΔJKL|||ΔPMN.

Determine the length of PN.

Answer: PN= units
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

If two triangles are similar, then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

STEP: Match up the sides and angles
[−2 points ⇒ 0 / 2 points left]

If two triangles are similar then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

We know that the triangles are similar, so their sides will be in proportion. We use this fact to determine the length of PN.

First we match up the sides. In the statement ΔJKL|||ΔPMN, JK and PM are the first two letters, so their sides must match up. Similarly, KL and MN match up because their letters came last. Finally, JL and PN must match up, because they are the only sides left.

Since JK and PM match up, we can work out the proportionality constant:

k=PMJKk=4928k=1.75

This is the number by which all sides in ΔJKL should by multiplied in order to get the matching sides in ΔPMN, because 28×1.75=49.

This is only true because we were told that ΔJKL|||ΔPMN. So, we use this as our reason.

So,

PN=44×1.75PN=77 units(ΔJKL|||ΔPMN)

Submit your answer as: and

ID is: 4289 Seed is: 2989

Consequences of similarity

In the diagram below, ΔXYZ|||ΔLJK.

Determine the length of LK.

Answer: LK= units
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

If two triangles are similar, then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

STEP: Match up the sides and angles
[−2 points ⇒ 0 / 2 points left]

If two triangles are similar then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

We know that the triangles are similar, so their sides will be in proportion. We use this fact to determine the length of LK.

First we match up the sides. In the statement ΔXYZ|||ΔLJK, XY and LJ are the first two letters, so their sides must match up. Similarly, YZ and JK match up because their letters came last. Finally, XZ and LK must match up, because they are the only sides left.

Since XY and LJ match up, we can work out the proportionality constant:

k=LJXYk=4518k=2.5

This is the number by which all sides in ΔXYZ should by multiplied in order to get the matching sides in ΔLJK, because 18×2.5=45.

This is only true because we were told that ΔXYZ|||ΔLJK. So, we use this as our reason.

So,

LK=20×2.5LK=50 units(ΔXYZ|||ΔLJK)

Submit your answer as: and

ID is: 4289 Seed is: 2990

Consequences of similarity

In the diagram below, ΔABC|||ΔNMP.

Determine the length of NP.

Answer: NP= units
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

If two triangles are similar, then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

STEP: Match up the sides and angles
[−2 points ⇒ 0 / 2 points left]

If two triangles are similar then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

We know that the triangles are similar, so their sides will be in proportion. We use this fact to determine the length of NP.

First we match up the sides. In the statement ΔABC|||ΔNMP, AB and NM are the first two letters, so their sides must match up. Similarly, BC and MP match up because their letters came last. Finally, AC and NP must match up, because they are the only sides left.

Since AB and NM match up, we can work out the proportionality constant:

k=NMABk=9040k=2.25

This is the number by which all sides in ΔABC should by multiplied in order to get the matching sides in ΔNMP, because 40×2.25=90.

This is only true because we were told that ΔABC|||ΔNMP. So, we use this as our reason.

So,

NP=32×2.25NP=72 units(ΔABC|||ΔNMP)

Submit your answer as: and

ID is: 4302 Seed is: 7041

Identify similar triangles

In the diagram below, STTR and SRQP. Also, R^=x, RQ=7, QT=4, and PR=6.

  1. Ekene needs to prove that ΔSTR|||ΔQPR. He has already started, and his incomplete proof is written below:

    In ΔSTR and ΔQPR:

    1. R^ is common
    2. ST^R=QP^R=90° (given)
      ...

    How should Ekene complete the proof? Choose the best option.

    Answer:
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Which can you use here?


    STEP: Choose the correct proof
    [−3 points ⇒ 0 / 3 points left]
    NOTE:

    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Ekene has already proved two pairs of angles equal for us, and we need to prove the third pair. To do so, it is helpful to look at each triangle separately.

    Looking at ΔSTR, we can write S^ in terms of x:

    S^+x+90°=180°(sum of s in Δ)S^+x+90°90°=180°90°S^+x=90°S^+xx=90°xS^=90°x

    In the same way in ΔQPR, we can write RQ^P in terms of x.

    RQ^P+x+90°=180°(sum of s in Δ)RQ^P=90°x
    NOTE: It is true that TQ^P is equal to 90°+x. But, this is not an angle in either triangle, so it does not help us to prove that ΔSTR|||ΔQPR.

    The full proof is shown below:

    In ΔSTR and ΔQPR:

    1. R^ is common
    2. ST^R=QP^R=90° (given)
    3. S^=90°x (sum of s in Δ)
      Also, RQ^P=90°x (sum of s in Δ)
      S^=RQ^P (both equal to 90°x)

    ΔSTR|||ΔQPR (equiangular Δs)


    Submit your answer as:
  2. Hence, determine the length of SP.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Give your answer as a fraction, where necessary.
    Answer: SP=
    one-of
    type(numeric.abserror(0.01))
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects SP to the sides that you already know. Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
    [−2 points ⇒ 2 / 4 points left]

    We are trying to find SP. Even though SP is not a side in the similar triangles, it is part of side SR:

    SR=SP+PR

    To make the algebra easier, let SP=y.

    Now we have

    SR=y+6

    The similarity is easier to see if we separate the triangles.

    We can see from the similarity statement that SR matches with QR: ΔSTR|||ΔQPR.

    We were given the lengths of RQ and QT. Since TR=RQ+QT, we know that TR=11. So we know the length of TR and PR, which match up: ΔSTR|||ΔQPR

    So, dividing the sides of the big triangle by the sides of the small triangle, we get:

    SRQR=TRPR(ΔSTR|||ΔQPR)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Solve the equation using inverse operations
    [−2 points ⇒ 0 / 4 points left]

    First we need to substitute in the values from the diagram. As long as there are only numbers and y-terms, we will be able to solve the equation.

    y+67=116y+67×7=116×7y+6=776y+66=7766y=416

    Submit your answer as: and

ID is: 4302 Seed is: 6366

Identify similar triangles

In the diagram below, RSST and RTSQ. Also, T^=x, ST=8, and QT=4.

  1. Chibueze needs to prove that ΔRST|||ΔSQT. He has already started, and his incomplete proof is written below:

    In ΔRST and ΔSQT:

    1. T^ is common
    2. RS^T=SQ^T=90° (given)
      ...

    How should Chibueze complete the proof? Choose the best option.

    Answer:
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Which can you use here?


    STEP: Choose the correct proof
    [−3 points ⇒ 0 / 3 points left]
    NOTE:

    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Chibueze has already proved two pairs of angles equal for us, and we need to prove the third pair. To do so, it is helpful to look at each triangle separately.

    Looking at ΔRST, we can write R^ in terms of x:

    R^+x+90°=180°(sum of s in Δ)R^+x+90°90°=180°90°R^+x=90°R^+xx=90°xR^=90°x

    In the same way in ΔSQT, we can write TS^Q in terms of x.

    TS^Q+x+90°=180°(sum of s in Δ)TS^Q=90°x
    NOTE: We cannot assume that QS cuts RS^T in half. TS^Q might be equal to 45°, or it might be something else. We must only use the information that we have been given.

    The full proof is shown below:

    In ΔRST and ΔSQT:

    1. T^ is common
    2. RS^T=SQ^T=90° (given)
    3. R^=90°x (sum of s in Δ)
      Also, TS^Q=90°x (sum of s in Δ)
      R^=TS^Q (both equal to 90°x)

    ΔRST|||ΔSQT (equiangular Δs)


    Submit your answer as:
  2. Hence, determine the length of RQ.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Give your answer as a fraction, where necessary.
    Answer: RQ=
    one-of
    type(numeric.abserror(0.01))
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects RQ to the sides that you already know. Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
    [−2 points ⇒ 2 / 4 points left]

    We are trying to find RQ. Even though RQ is not a side in the similar triangles, it is part of side RT:

    RT=RQ+QT

    To make the algebra easier, let RQ=y.

    Now we have

    RT=y+4

    The similarity is easier to see if we separate the triangles.

    We can see from the similarity statement that RT matches with ST: ΔRST|||ΔSQT.

    We know the lengths of ST and QT. They are a matching pair of sides: (ΔRST|||ΔSQT).

    So, dividing the sides of the big triangle by the sides of the small triangle, we get:

    RTST=STQT(ΔRST|||ΔSQT)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Solve the equation using inverse operations
    [−2 points ⇒ 0 / 4 points left]

    First we need to substitute in the values from the diagram. As long as there are only numbers and y-terms, we will be able to solve the equation.

    y+48=84y+48×8=84×8y+4=644y+44=6444y=12

    Submit your answer as: and

ID is: 4302 Seed is: 4622

Identify similar triangles

In the diagram below, TSSQ and TQPR. Also, Q^=x, QP=6, PS=4, and RQ=5.

  1. Justine needs to prove that ΔTSQ|||ΔPRQ. She has already started, and her incomplete proof is written below:

    In ΔTSQ and ΔPRQ:

    1. Q^ is common
    2. TS^Q=PR^Q=90° (given)
      ...

    How should Justine complete the proof? Choose the best option.

    Answer:
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Which can you use here?


    STEP: Choose the correct proof
    [−3 points ⇒ 0 / 3 points left]
    NOTE:

    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Justine has already proved two pairs of angles equal for us, and we need to prove the third pair. To do so, it is helpful to look at each triangle separately.

    Looking at ΔTSQ, we can write T^ in terms of x:

    T^+x+90°=180°(sum of s in Δ)T^+x+90°90°=180°90°T^+x=90°T^+xx=90°xT^=90°x

    In the same way in ΔPRQ, we can write QP^R in terms of x.

    QP^R+x+90°=180°(sum of s in Δ)QP^R=90°x
    NOTE: It is true that SP^R is equal to 90°+x. But, this is not an angle in either triangle, so it does not help us to prove that ΔTSQ|||ΔPRQ.

    The full proof is shown below:

    In ΔTSQ and ΔPRQ:

    1. Q^ is common
    2. TS^Q=PR^Q=90° (given)
    3. T^=90°x (sum of s in Δ)
      Also, QP^R=90°x (sum of s in Δ)
      T^=QP^R (both equal to 90°x)

    ΔTSQ|||ΔPRQ (equiangular Δs)


    Submit your answer as:
  2. Hence, determine the length of TR.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Give your answer as a fraction, where necessary.
    Answer: TR=
    one-of
    type(numeric.abserror(0.01))
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects TR to the sides that you already know. Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
    [−2 points ⇒ 2 / 4 points left]

    We are trying to find TR. Even though TR is not a side in the similar triangles, it is part of side TQ:

    TQ=TR+RQ

    To make the algebra easier, let TR=y.

    Now we have

    TQ=y+5

    The similarity is easier to see if we separate the triangles.

    We can see from the similarity statement that TQ matches with PQ: ΔTSQ|||ΔPRQ.

    We were given the lengths of QP and PS. Since SQ=QP+PS, we know that SQ=10. So we know the length of SQ and RQ, which match up: ΔTSQ|||ΔPRQ

    So, dividing the sides of the big triangle by the sides of the small triangle, we get:

    TQPQ=SQRQ(ΔTSQ|||ΔPRQ)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Solve the equation using inverse operations
    [−2 points ⇒ 0 / 4 points left]

    First we need to substitute in the values from the diagram. As long as there are only numbers and y-terms, we will be able to solve the equation.

    y+56=105y+56×6=105×6y+5=605y+55=6055y=7

    Submit your answer as: and

ID is: 4258 Seed is: 3184

Definition: Similarity

Two triangles are similar if:

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

What do we mean in maths when we say that two triangles are similar?


STEP: Identify the definition of similarity
[−2 points ⇒ 0 / 2 points left]

We say that two triangles are similar if they are exactly the same shape, but one might be bigger or smaller than the other one. Another way of saying this is that one is an enlargement of the other one.

For example, ΔPQR and ΔXYZ below are the same shape. But, they are different sizes.

The sides of similar triangles are in proportion. That means that all of the sides in one triangle have been multiplied by the same number to get the sides in the second triangle. All of the sides in ΔPQR have been multiplied by 2 to get ΔXYZ.

Similar triangles are the same shape, so they always have the same angles. We can see the matching angles in ΔPQR and ΔXYZ.

NOTE: In normal English, we say two things are similar if they are almost the same. But, in Maths, it is not precise enough to say that similar shapes are approximately the same. In Maths, similarity has a very specific meaning.

Two triangles are similar if either

  • their matching sides are in proportion, or
  • their matching angles are equal.
NOTE: In many polygons, we have to prove that the matching sides are in proportion and the matching angles are equal, if we want to be sure that the shapes are similar. But, in triangles, we only need to prove that one of them is true. This is a special property of triangles, which you will learn more about in grade 12.

Submit your answer as:

ID is: 4258 Seed is: 7198

Definition: Similarity

Two triangles are similar if:

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

What do we mean in maths when we say that two triangles are similar?


STEP: Identify the definition of similarity
[−2 points ⇒ 0 / 2 points left]

We say that two triangles are similar if they are exactly the same shape, but one might be bigger or smaller than the other one. Another way of saying this is that one is an enlargement of the other one.

For example, ΔPQR and ΔXYZ below are the same shape. But, they are different sizes.

The sides of similar triangles are in proportion. That means that all of the sides in one triangle have been multiplied by the same number to get the sides in the second triangle. All of the sides in ΔPQR have been multiplied by 0.5 to get ΔXYZ.

Similar triangles are the same shape, so they always have the same angles. We can see the matching angles in ΔPQR and ΔXYZ.

NOTE: In normal English, we say two things are similar if they are almost the same. But, in Maths, it is not precise enough to say that similar shapes are approximately the same. In Maths, similarity has a very specific meaning.

Two triangles are similar if either

  • their matching sides are in proportion, or
  • their matching angles are equal.
NOTE: In many polygons, we have to prove that the matching sides are in proportion and the matching angles are equal, if we want to be sure that the shapes are similar. But, in triangles, we only need to prove that one of them is true. This is a special property of triangles, which you will learn more about in grade 12.

Submit your answer as:

ID is: 4258 Seed is: 3679

Definition: Similarity

Two triangles are similar if:

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

What do we mean in maths when we say that two triangles are similar?


STEP: Identify the definition of similarity
[−2 points ⇒ 0 / 2 points left]

We say that two triangles are similar if they are exactly the same shape, but one might be bigger or smaller than the other one. Another way of saying this is that one is an enlargement of the other one.

For example, ΔABC and ΔDEF below are the same shape. But, they are different sizes.

The sides of similar triangles are in proportion. That means that all of the sides in one triangle have been multiplied by the same number to get the sides in the second triangle. All of the sides in ΔABC have been multiplied by 2 to get ΔDEF.

Similar triangles are the same shape, so they always have the same angles. We can see the matching angles in ΔABC and ΔDEF.

NOTE: In normal English, we say two things are similar if they are almost the same. But, in Maths, it is not precise enough to say that similar shapes are approximately the same. In Maths, similarity has a very specific meaning.

Two triangles are similar if either

  • their matching sides are in proportion, or
  • their matching angles are equal.
NOTE: In many polygons, we have to prove that the matching sides are in proportion and the matching angles are equal, if we want to be sure that the shapes are similar. But, in triangles, we only need to prove that one of them is true. This is a special property of triangles, which you will learn more about in grade 12.

Submit your answer as:

ID is: 3091 Seed is: 7688

Using the proportionality of similar polygons

In the diagram below, JKNP|||LMPQ. Compute the value of f. The diagram is not necessarily drawn to scale.

Answer: f=
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Similar shapes have proportional lengths. Write a proportion for the sides of the polygons and then solve the equation for f.


STEP: Identify corresponding sides and write a proportion for these sides
[−1 point ⇒ 2 / 3 points left]

We have a diagram with quadrilaterals, and we need to figure out the value of the variable f. The question tells us that the quadrilaterals are similar, which means that the sides of the shapes are proportional. Therefore, we can write a proportion for the sides of the polygons and solve it for f.

There are a number of different proportions that we can write to summarise the lengths of the sides in the diagram: we can write a proportion for any pairs of corresponding sides. Here are two proportions which are accurate (but there are others):

JKLQ=NPMPJKNP=LQMP

STEP: Solve the equation for f
[−2 points ⇒ 0 / 3 points left]

We will use the first proportion from above to find the value of f. If you use a different correct proportion and solve it without any mistakes, you will get the correct answer for f (and full marks).

Remember:

MP=NP+MN=f+3+2=f+5

So now we can calculate f:

JKLQ=NPMP2f2f+4=f+3f+5(2f)(f+5)=(f+3)(2f+4)2f2+10f=2f2+2f+12cancel2f2 terms8f=12f=128f=32

Therefore, the value of the variable is f=32.


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ID is: 3091 Seed is: 2401

Using the proportionality of similar polygons

Polygons PQT and PRS are similar. Compute the value of f. The diagram is not necessarily drawn to scale.

Answer: f=
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Similar shapes have proportional lengths. Write a proportion for the sides of the polygons and then solve the equation for f.


STEP: Identify corresponding sides and write a proportion for these sides
[−1 point ⇒ 2 / 3 points left]

We have a diagram with triangles, and we need to figure out the value of the variable f. The question tells us that the triangles are similar, which means that the sides of the shapes are proportional. Therefore, we can write a proportion for the sides of the polygons and solve it for f.

There are a number of different proportions that we can write to summarise the lengths of the sides in the diagram: we can write a proportion for any pairs of corresponding sides. Here are two proportions which are accurate (but there are others):

PQQR=PTSTPQPR=PTPS

STEP: Solve the equation for f
[−2 points ⇒ 0 / 3 points left]

We will use the first proportion from above to find the value of f. If you use a different correct proportion and solve it without any mistakes, you will get the correct answer for f (and full marks).

PQQR=PTST3f12=f+63(3f)(3)=(f+6)(12)9f=12f+7221f=72f=7221f=247

Therefore, the value of the variable is f=247.


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ID is: 3091 Seed is: 5555

Using the proportionality of similar polygons

In the diagram below, ABEF|||CDFG. Compute the value of y. The diagram is not necessarily drawn to scale.

Answer: y=
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Similar shapes have proportional lengths. Write a proportion for the sides of the polygons and then solve the equation for y.


STEP: Identify corresponding sides and write a proportion for these sides
[−1 point ⇒ 2 / 3 points left]

We have a diagram with quadrilaterals, and we need to figure out the value of the variable y. The question tells us that the quadrilaterals are similar, which means that the sides of the shapes are proportional. Therefore, we can write a proportion for the sides of the polygons and solve it for y.

There are a number of different proportions that we can write to summarise the lengths of the sides in the diagram: we can write a proportion for any pairs of corresponding sides. Here are two proportions which are accurate (but there are others):

ABCG=EFDFABEF=CGDF

STEP: Solve the equation for y
[−2 points ⇒ 0 / 3 points left]

We will use the first proportion from above to find the value of y. If you use a different correct proportion and solve it without any mistakes, you will get the correct answer for y (and full marks).

Remember:

DF=EF+DE=2y+4+16=2y+20

So now we can calculate y:

ABCG=EFDFyy+4=2y+42y+20(y)(2y+20)=(2y+4)(y+4)2y2+20y=2y24y+16cancel2y2 terms24y=16y=1624y=23

Therefore, the value of the variable is y=23.


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ID is: 4301 Seed is: 3456

Given similar triangles, calculate a side

In the diagram below, ΔLNJ|||ΔKNM. Also, JL=7, MK=3, MJ=6, and NM=b .

Determine the value of b, giving a reason for your answer.

INSTRUCTION: Give your answer as a fraction, where necessary.
Answer: b= units
one-of
type(numeric.abserror(0.01))
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects b to the sides that you already know. You may have b more than once in your equation!


STEP: Use the similarity statement to choose the correct pair of ratios
[−2 points ⇒ 2 / 4 points left]

If two triangles are similar, their sides are in proportion. So, when you divide the sides of one triangle with the matching sides in the other triangle, you always get the same number.

TIP: Use the similarity statement to choose the correct pair of ratios by making pairs of matching sides.

We are trying to find b. We can see that b is equal to NM. We can also see that:

NJ=b+6

The similarity is easier to see if we separate the triangles.

We can see from the similarity statement that NJ matches with NM: ΔLNJ|||ΔKNM.

We know the lengths of LJ and KM. They are a matching pair of sides: ΔLNJ|||ΔKNM.

So, dividing the sides in the big triangle by those in the small triangle, we get

NJNM=LJKM(ΔLNJ|||ΔKNM)
NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

STEP: Solve the equation using inverse operations
[−2 points ⇒ 0 / 4 points left]

First we need to substitute in the values from the diagram. As long as there are only numbers and b-terms, we will be able to solve the equation.

b+6b=73b+6b×b=73×bb+6=7b3(b+6)×3=7b3×33b+18=7b3b+183b=7b3b18=4bb=92

Submit your answer as: and

ID is: 4301 Seed is: 1050

Given similar triangles, calculate a side

In the diagram below, ΔPST|||ΔRQT. Also, SP=11, QR=6, TQ=5, and QS=b .

Determine the value of b, giving a reason for your answer.

INSTRUCTION: Give your answer as a fraction, where necessary.
Answer: b= units
one-of
type(numeric.abserror(0.01))
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects b to the sides that you already know. You may have b more than once in your equation!


STEP: Use the similarity statement to choose the correct pair of ratios
[−2 points ⇒ 2 / 4 points left]

If two triangles are similar, their sides are in proportion. So, when you divide the sides of one triangle with the matching sides in the other triangle, you always get the same number.

TIP: Use the similarity statement to choose the correct pair of ratios by making pairs of matching sides.

We are trying to find b. Even though b is not a side in a triangle, it is part of side TS:

TS=b+5

The similarity is easier to see if we separate the triangles.

We can see from the similarity statement that ST matches with QT: ΔPST|||ΔRQT.

We know the lengths of PS and RQ. They are a matching pair of sides: ΔPST|||ΔRQT.

So, dividing the sides in the big triangle by those in the small triangle, we get

STQT=PSRQ(ΔPST|||ΔRQT)
NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

STEP: Solve the equation using inverse operations
[−2 points ⇒ 0 / 4 points left]

First we need to substitute in the values from the diagram. As long as there are only numbers and b-terms, we will be able to solve the equation.

b+55=116b+55×5=116×5b+5=556b+55=5565b=256

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ID is: 4301 Seed is: 4547

Given similar triangles, calculate a side

In the diagram below, ΔCBD|||ΔCEA. Also, BD=8, EA=3, EB=7, and CE=y .

Determine the value of y, giving a reason for your answer.

INSTRUCTION: Give your answer as a fraction, where necessary.
Answer: y= units
one-of
type(numeric.abserror(0.01))
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects y to the sides that you already know. You may have y more than once in your equation!


STEP: Use the similarity statement to choose the correct pair of ratios
[−2 points ⇒ 2 / 4 points left]

If two triangles are similar, their sides are in proportion. So, when you divide the sides of one triangle with the matching sides in the other triangle, you always get the same number.

TIP: Use the similarity statement to choose the correct pair of ratios by making pairs of matching sides.

We are trying to find y. We can see that y is equal to CE. We can also see that:

CB=y+7

The similarity is easier to see if we separate the triangles.

We can see from the similarity statement that CB matches with CE: ΔCBD|||ΔCEA.

We know the lengths of BD and EA. They are a matching pair of sides: ΔCBD|||ΔCEA.

So, dividing the sides in the big triangle by those in the small triangle, we get

CBCE=BDEA(ΔCBD|||ΔCEA)
NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

STEP: Solve the equation using inverse operations
[−2 points ⇒ 0 / 4 points left]

First we need to substitute in the values from the diagram. As long as there are only numbers and y-terms, we will be able to solve the equation.

y+7y=83y+7y×y=83×yy+7=8y3(y+7)×3=8y3×33y+21=8y3y+213y=8y3y21=5yy=215

Submit your answer as: and

ID is: 4304 Seed is: 1103

Given similar triangles, calculate a side

  1. In the diagram below, ΔLJK|||ΔLMJ. Also, MJ=24, LJ=26, and LM=10.

    Olatunji has been asked to determine the length of JK. He knows that in order to calculate JK, he first needs to match up the sides of the two similar triangles.

    Help Olatunji to match up the sides in the similar triangles.

    Answer:
    Side in ΔLJK Matching side in ΔLMJ
    LJ
    LK
    JK
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Use the similarity statement to choose the correct pairs of sides.


    STEP: Use the similarity statement to match the sides
    [−2 points ⇒ 0 / 2 points left]

    Since ΔLJK|||ΔLMJ, we can see that LJ matches with LM. These are the first two letters in the similarity statement.

    Also using ΔLJK|||ΔLMJ, we can see that LK matches with LJ. These are the first and last letters in the similarity statement.

    Finally, we can use ΔLJK|||ΔLMJ to work out that JK matches with MJ. These are the second and third letters in the similarity statement.

    Visually we can see the matching sides of the triangles:


    Submit your answer as: andand
  2. Determine the value of JK.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Round your answer to two decimal places, if necessary.
    Answer: JK= units
    one-of
    type(numeric.abserror(0.01))
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similaity statement to choose the correct pair of ratios
    [−2 points ⇒ 1 / 3 points left]

    If two triangles are similar, then their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for JK. So, start by writing JK in the numerator:

    JK=

    Since ΔLJK|||ΔLMJ, we can see that JK matches with MJ.

    So we have

    JKMJ=

    We were also given information about LJ and LM. Since ΔLJK|||ΔLMJ, we can see that LJ matches with LM.

    So,

    JKMJ=LJLM(ΔLJK|||ΔLMJ)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Substitute in the lengths of the sides and solve the equation
    [−1 point ⇒ 0 / 3 points left]
    JK24=2610JK=62.4 units

    Submit your answer as: and

ID is: 4304 Seed is: 5964

Given similar triangles, calculate a side

  1. In the diagram below, ΔLJM|||ΔLKJ. Also, LJ=15 and LK=9.

    Abdulai has been asked to determine the length of LM. He knows that in order to calculate LM, he first needs to match up the sides of the two similar triangles.

    Help Abdulai to match up the sides in the similar triangles.

    Answer:
    Side in ΔLJM Matching side in ΔLKJ
    LJ
    LM
    JM
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Use the similarity statement to choose the correct pairs of sides.


    STEP: Use the similarity statement to match the sides
    [−2 points ⇒ 0 / 2 points left]

    Since ΔLJM|||ΔLKJ, we can see that LJ matches with LK. These are the first two letters in the similarity statement.

    Also using ΔLJM|||ΔLKJ, we can see that LM matches with LJ. These are the first and last letters in the similarity statement.

    Finally, we can use ΔLJM|||ΔLKJ to work out that JM matches with KJ. These are the second and third letters in the similarity statement.

    Visually we can see the matching sides of the triangles:


    Submit your answer as: andand
  2. Determine the value of LM.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Round your answer to two decimal places, if necessary.
    Answer: LM= units
    one-of
    type(numeric.abserror(0.01))
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similaity statement to choose the correct pair of ratios
    [−2 points ⇒ 1 / 3 points left]

    If two triangles are similar, then their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for LM. So, start by writing LM in the numerator:

    LM=

    Since ΔLJM|||ΔLKJ, we can see that LM matches with LJ.

    So we have

    LMLJ=

    We were also given information about LJ and LK. Since ΔLJM|||ΔLKJ, we can see that LJ matches with LK.

    So,

    LMLJ=LJLK(ΔLJM|||ΔLKJ)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Substitute in the lengths of the sides and solve the equation
    [−1 point ⇒ 0 / 3 points left]
    LM15=159LM=25 units

    Submit your answer as: and

ID is: 4304 Seed is: 7470

Given similar triangles, calculate a side

  1. In the diagram below, ΔRSP|||ΔRQS. Also, QS=12, RS=15, and RQ=9.

    Anita has been asked to determine the length of SP. She knows that in order to calculate SP, she first needs to match up the sides of the two similar triangles.

    Help Anita to match up the sides in the similar triangles.

    Answer:
    Side in ΔRSP Matching side in ΔRQS
    RS
    RP
    SP
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Use the similarity statement to choose the correct pairs of sides.


    STEP: Use the similarity statement to match the sides
    [−2 points ⇒ 0 / 2 points left]

    Since ΔRSP|||ΔRQS, we can see that RS matches with RQ. These are the first two letters in the similarity statement.

    Also using ΔRSP|||ΔRQS, we can see that RP matches with RS. These are the first and last letters in the similarity statement.

    Finally, we can use ΔRSP|||ΔRQS to work out that SP matches with QS. These are the second and third letters in the similarity statement.

    Visually we can see the matching sides of the triangles:


    Submit your answer as: andand
  2. Determine the value of SP.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Round your answer to two decimal places, if necessary.
    Answer: SP= units
    one-of
    type(numeric.abserror(0.01))
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similaity statement to choose the correct pair of ratios
    [−2 points ⇒ 1 / 3 points left]

    If two triangles are similar, then their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for SP. So, start by writing SP in the numerator:

    SP=

    Since ΔRSP|||ΔRQS, we can see that SP matches with QS.

    So we have

    SPQS=

    We were also given information about RS and RQ. Since ΔRSP|||ΔRQS, we can see that RS matches with RQ.

    So,

    SPQS=RSRQ(ΔRSP|||ΔRQS)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Substitute in the lengths of the sides and solve the equation
    [−1 point ⇒ 0 / 3 points left]
    SP12=159SP=20 units

    Submit your answer as: and

ID is: 4261 Seed is: 5286

Similarity and congruency notation

Consider the following triangles:

Answer:

Which of the following best describes the relationship between ΔPQR and ΔXYZ?

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The symbol is used to indicate that two triangles are congruent. The symbol ||| is used to indicate that two triangles are similar.


STEP: Identify the triangle relationship
[−2 points ⇒ 0 / 2 points left]

Two triangles are similar if they are the same shape, but they can be a different size. The symbol for similarity is |||. There are two ways to know if triangles are similar:

  • if their sides are in proportion. This means that the sides in one triangle have been multiplied by the same number to get the sides in another triangle.
  • if all of their matching angles are equal.

Two triangles are congruent if they are exactly the same size and shape. The symbol for congruency is . There are four ways to know if triangles are congruent: SSS, SAS, SAA, or 90°HS.

TIP: When two triangles are equal in all respects, we say that they are congruent and use the symbol for congruency (). We never use the equals sign (=) to compare two triangles.

Two sides and an angle were given in both triangles. But, the angle was not between the two sides, so this is not a case for congruency. This is why we have to prove the included angle in the SAS congruency case. Anyway, the triangles do not look congruent: if we put one on top of the other, they would not match up.

This is also not a case for similarity because the triangles are not the same shape.


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Similarity and congruency notation

Consider the following triangles:

Answer:

Which of the following best describes the relationship between ΔDEF and ΔJKL?

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The symbol is used to indicate that two triangles are congruent. The symbol ||| is used to indicate that two triangles are similar.


STEP: Identify the triangle relationship
[−2 points ⇒ 0 / 2 points left]

Two triangles are similar if they are the same shape, but they can be a different size. The symbol for similarity is |||. There are two ways to know if triangles are similar:

  • if their sides are in proportion. This means that the sides in one triangle have been multiplied by the same number to get the sides in another triangle.
  • if all of their matching angles are equal.

Two triangles are congruent if they are exactly the same size and shape. The symbol for congruency is . There are four ways to know if triangles are congruent: SSS, SAS, SAA, or 90°HS.

TIP: When two triangles are equal in all respects, we say that they are congruent and use the symbol for congruency (). We never use the equals sign (=) to compare two triangles.

We were given three sides in both triangles. So, the triangles are congruent, and we write:

ΔDEFΔJKL (SSS)

NOTE: If two triangles are congruent, then they are always also similar. But, being congruent is more special that being similar, so the best way to describe them is to say that they are congruent.

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ID is: 4261 Seed is: 9067

Similarity and congruency notation

Consider the following triangles:

Answer:

Which of the following best describes the relationship between ΔABC and ΔDEF?

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The symbol is used to indicate that two triangles are congruent. The symbol ||| is used to indicate that two triangles are similar.


STEP: Identify the triangle relationship
[−2 points ⇒ 0 / 2 points left]

Two triangles are similar if they are the same shape, but they can be a different size. The symbol for similarity is |||. There are two ways to know if triangles are similar:

  • if their sides are in proportion. This means that the sides in one triangle have been multiplied by the same number to get the sides in another triangle.
  • if all of their matching angles are equal.

Two triangles are congruent if they are exactly the same size and shape. The symbol for congruency is . There are four ways to know if triangles are congruent: SSS, SAS, SAA, or 90°HS.

TIP: When two triangles are equal in all respects, we say that they are congruent and use the symbol for congruency (). We never use the equals sign (=) to compare two triangles.

The sides in ΔDEF are each 2 units longer than the sides in ΔABC. But, this does not mean the sides are in proportion. For the sides to be in proportion, we must multiply by the same number each time (not add). When we divide the pairs of sides in order of length, we do not get the same number each time:

ABDE=79=0.77777...BCEF=14.516.5=0.87878...CAFD=12.514.5=0.86206...

Even though these triangles look like they are approximately the same shape, they are not similar (or congruent).


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ID is: 4260 Seed is: 5866

Determine the proportionality factor between similar triangles

The diagram below shows two similar triangles:

All of the sides in ΔKJL been multiplied by the same number, k, to get the sides in ΔPMN. What is the value of k?

INSTRUCTION: Give your answer as an exact decimal or as a fraction.
Answer: k=
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Divide the longest side in ΔPMN by the longest side in ΔKJL.


STEP: Divide the longest side in ΔPMN by the longest side in ΔKJL
[−2 points ⇒ 0 / 2 points left]

We are looking for the number that each matching side in ΔKJL has been multiplied by, to get the matching side in ΔPMN.

NOTE: We call this number the proportionality constant, k.

The longest side in ΔKJL is 24. It must match up with the longest side in ΔPMN, which is 54. We will use these sides to determine the multiplier. Let the proportionality constant be k:

24×k=5424×k24=5424k=2.25
NOTE: In the last step, we divided the longest side in ΔPMN by the longest side in ΔKJL, to find k.

We can check that this value works for the other sides:

  • The shortest sides, KJ and PM: 12×2.25=27
  • The middle sides, JL and MN: 16×2.25=36

So, all the sides in ΔKJL have been multiplied by 2.25 to get the sides in ΔPMN. We call this number the proportionality constant.


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ID is: 4260 Seed is: 3237

Determine the proportionality factor between similar triangles

The diagram below shows two similar triangles:

All of the sides in ΔJLK been multiplied by the same number, k, to get the sides in ΔPMN. What is the value of k?

INSTRUCTION: Give your answer as an exact decimal or as a fraction.
Answer: k=
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Divide the longest side in ΔPMN by the longest side in ΔJLK.


STEP: Divide the longest side in ΔPMN by the longest side in ΔJLK
[−2 points ⇒ 0 / 2 points left]

We are looking for the number that each matching side in ΔJLK has been multiplied by, to get the matching side in ΔPMN.

NOTE: We call this number the proportionality constant, k.

The longest side in ΔJLK is 14. It must match up with the longest side in ΔPMN, which is 21. We will use these sides to determine the multiplier. Let the proportionality constant be k:

14×k=2114×k14=2114k=1.5
NOTE: In the last step, we divided the longest side in ΔPMN by the longest side in ΔJLK, to find k.

We can check that this value works for the other sides:

  • The shortest sides, JL and PM: 10×1.5=15
  • The middle sides, LK and MN: 13×1.5=19.5

So, all the sides in ΔJLK have been multiplied by 1.5 to get the sides in ΔPMN. We call this number the proportionality constant.


Submit your answer as:

ID is: 4260 Seed is: 9390

Determine the proportionality factor between similar triangles

The diagram below shows two similar triangles:

All of the sides in ΔKJL been multiplied by the same number, k, to get the sides in ΔNPM. What is the value of k?

INSTRUCTION: Give your answer as an exact decimal or as a fraction.
Answer: k=
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Divide the longest side in ΔNPM by the longest side in ΔKJL.


STEP: Divide the longest side in ΔNPM by the longest side in ΔKJL
[−2 points ⇒ 0 / 2 points left]

We are looking for the number that each matching side in ΔKJL has been multiplied by, to get the matching side in ΔNPM.

NOTE: We call this number the proportionality constant, k.

The longest side in ΔKJL is 9. It must match up with the longest side in ΔNPM, which is 22.5. We will use these sides to determine the multiplier. Let the proportionality constant be k:

9×k=22.59×k9=22.59k=2.5
NOTE: In the last step, we divided the longest side in ΔNPM by the longest side in ΔKJL, to find k.

We can check that this value works for the other sides:

  • The shortest sides, KJ and NP: 5×2.5=12.5
  • The middle sides, JL and PM: 8×2.5=20

So, all the sides in ΔKJL have been multiplied by 2.5 to get the sides in ΔNPM. We call this number the proportionality constant.


Submit your answer as:

ID is: 4294 Seed is: 1891

Similar, congruent, or neither?

Consider the following polygons:

Are the polygons similar, congruent, or neither? Choose the most correct description.

Answer:

The two polygons are:

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]
  • Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.
  • Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal. If we cut out two congruent polygons, one could be placed exactly on top of the other.

STEP: Identify if the polygons are similar, congruent, or neither
[−1 point ⇒ 0 / 1 points left]
Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal in size and all of their matching sides are in proportion.
Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal.

The two polygons do not have to be in the same orientation. This means that they can face different directions.

These two quadrilaterals look approximately the same. But, their angles are not exactly equal. So, they cannot be either similar or congruent.

NOTE: In normal English similar means the same in some ways. But the mathematical meaning of the word similar is very specific!

Submit your answer as:

ID is: 4294 Seed is: 1206

Similar, congruent, or neither?

Consider the following polygons:

Are the polygons similar, congruent, or neither? Choose the most correct description.

Answer:

The two polygons are:

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]
  • Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.
  • Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal. If we cut out two congruent polygons, one could be placed exactly on top of the other.

STEP: Identify if the polygons are similar, congruent, or neither
[−1 point ⇒ 0 / 1 points left]
Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal in size and all of their matching sides are in proportion.
Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal.

The two polygons do not have to be in the same orientation. This means that they can face different directions.

The matching sides and angles in the two polygons are exactly equal.

So, the two polygons are congruent.

NOTE: Congruent polygons are always also similar to each other. But if two polygons are exactly equal, it is more correct to call them congruent.

Submit your answer as:

ID is: 4294 Seed is: 7911

Similar, congruent, or neither?

Consider the following polygons:

Are the polygons similar, congruent, or neither? Choose the most correct description.

Answer:

The two polygons are:

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]
  • Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.
  • Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal. If we cut out two congruent polygons, one could be placed exactly on top of the other.

STEP: Identify if the polygons are similar, congruent, or neither
[−1 point ⇒ 0 / 1 points left]
Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal in size and all of their matching sides are in proportion.
Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal.

The two polygons do not have to be in the same orientation. This means that they can face different directions.

The matching sides and angles in the two polygons are exactly equal.

So, the two polygons are congruent.

NOTE: Congruent polygons are always also similar to each other. But if two polygons are exactly equal, it is more correct to call them congruent.

Submit your answer as:

ID is: 4256 Seed is: 3792

Identify similar triangles

ΔPRQ is drawn below.

Which of the following triangles is definitely similar to ΔPRQ? Give a reason for the similarity.

Answer:

ΔPRQ|||Δ

string
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Triangles are similar if either their:

  • matching sides are in proportion (sides of Δ in prop), or
  • matching angles are equal (equiangular Δs).

STEP: Identify the similar triangles
[−1 point ⇒ 1 / 2 points left]

Triangles are similar if either their:

  • sides are in proportion (sides of Δ in prop), or
  • angles are equal (equiangular Δs).

We were given three pairs of equal angles in the two triangles, so they are similar.


STEP: Match up the triangle vertices
[−1 point ⇒ 0 / 2 points left]

We must name the triangles in the order that their angles match up.

We can see that the angle at P is equal to the angle at E. So, Point P matches with Point E.

In the same way:

  • Point R matches with D, and
  • Point Q matches with F.

So,

ΔPRQ|||ΔEDF (equiangular Δs)

NOTE: Equiangular comes from the equal angles in both triangles. It looks similar to equilateral, which means equal lines. But, the two words do not mean the same thing.

Submit your answer as: and

ID is: 4256 Seed is: 1062

Identify similar triangles

ΔCAB is drawn below.

Which of the following triangles is definitely similar to ΔCAB? Give a reason for the similarity.

Answer:

ΔCAB|||Δ

string
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Triangles are similar if either their:

  • matching sides are in proportion (sides of Δ in prop), or
  • matching angles are equal (equiangular Δs).

STEP: Identify the similar triangles
[−1 point ⇒ 1 / 2 points left]

Triangles are similar if either their:

  • sides are in proportion (sides of Δ in prop), or
  • angles are equal (equiangular Δs).

We were given three pairs of equal angles in the two triangles, so they are similar.


STEP: Match up the triangle vertices
[−1 point ⇒ 0 / 2 points left]

We must name the triangles in the order that their angles match up.

We can see that the angle at C is equal to the angle at D. So, Point C matches with Point D.

In the same way:

  • Point A matches with E, and
  • Point B matches with F.

So,

ΔCAB|||ΔDEF (equiangular Δs)

NOTE: Equiangular comes from the equal angles in both triangles. It looks similar to equilateral, which means equal lines. But, the two words do not mean the same thing.

Submit your answer as: and

ID is: 4256 Seed is: 7399

Identify similar triangles

ΔQRP is drawn below.

Which of the following triangles is definitely similar to ΔQRP? Give a reason for the similarity.

Answer:

ΔQRP|||Δ

string
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Triangles are similar if either their:

  • matching sides are in proportion (sides of Δ in prop), or
  • matching angles are equal (equiangular Δs).

STEP: Identify the similar triangles
[−1 point ⇒ 1 / 2 points left]

Triangles are similar if either their:

  • sides are in proportion (sides of Δ in prop), or
  • angles are equal (equiangular Δs).

We were given three pairs of equal angles in the two triangles, so they are similar.


STEP: Match up the triangle vertices
[−1 point ⇒ 0 / 2 points left]

We must name the triangles in the order that their angles match up.

We can see that the angle at Q is equal to the angle at F. So, Point Q matches with Point F.

In the same way:

  • Point R matches with E, and
  • Point P matches with D.

So,

ΔQRP|||ΔFED (equiangular Δs)

NOTE: Equiangular comes from the equal angles in both triangles. It looks similar to equilateral, which means equal lines. But, the two words do not mean the same thing.

Submit your answer as: and

ID is: 4285 Seed is: 262

Prove simple similarity with sides

Consider the following triangles:

Prove that ΔBAC|||ΔEDF.

INSTRUCTION: Give your answer as a simplified fraction or decimal.
Answer:

In ΔBAC and ΔEDF:

  1. ED÷ =
  2. DF÷ =
  3. EF÷ =

ΔBAC|||ΔEDF (sides of Δ in prop)

numeric
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

Which of these strategies is more appropriate for this question?


STEP: Match up sides and find the proportionality constant
[−3 points ⇒ 1 / 4 points left]

There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

In this question, we have more information about the sides of the triangles than about the angles. So, we will prove that all of the sides are in proportion.

We must show that each side must be multiplied by the same number to make its matching side in the other triangle.

Firstly, we must match up the sides:

  • The longest sides (ED and BA) match up.
  • The shortest sides (EF and BC) match up.
  • FD and AC must match up because they are the only sides left.

Now we will work out what each side in ΔBAC should be multiplied by:

EDBA=2016=1.25DFAC=1512=1.25EFBC=108=1.25

This means that any side in ΔBAC must be multiplied by 1.25 to make its matching side in ΔEDF.

For example, 16×1.25=20.

We call this number the proportionality constant.

If any one of these division sums had given us a different answer, the sides of the triangles would not have been in proportion. Then, the triangles would not be similar.

TIP: If your calculations tell you that the triangles are not similar in a question where you are asked to prove similarity, you probably made a mistake somewhere. Check your working out!

STEP: Prove similarity
[−1 point ⇒ 0 / 4 points left]

Now that we have matched up the sides, and worked out the proportionality constant, we just need to put the information into a formal similarity proof:

In ΔBAC and ΔEDF:

  1. ED÷BA = 2016 = 1.25
  2. DF÷AC = 1512 = 1.25
  3. EF÷BC = 108 = 1.25

ΔBAC|||ΔEDF (sides of Δ in prop)


Submit your answer as: andandandandand

ID is: 4285 Seed is: 2870

Prove simple similarity with sides

Consider the following triangles:

Prove that ΔQPR|||ΔXYZ.

INSTRUCTION: Give your answer as a simplified fraction or decimal.
Answer:

In ΔQPR and ΔXYZ:

  1. XY÷ =
  2. YZ÷ =
  3. XZ÷ =

ΔQPR|||ΔXYZ (sides of Δ in prop)

numeric
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

Which of these strategies is more appropriate for this question?


STEP: Match up sides and find the proportionality constant
[−3 points ⇒ 1 / 4 points left]

There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

In this question, we have more information about the sides of the triangles than about the angles. So, we will prove that all of the sides are in proportion.

We must show that each side must be multiplied by the same number to make its matching side in the other triangle.

Firstly, we must match up the sides:

  • The longest sides (XY and QP) match up.
  • The shortest sides (XZ and QR) match up.
  • ZY and PR must match up because they are the only sides left.

Now we will work out what each side in ΔQPR should be multiplied by:

XYQP=1812=1.5YZPR=1510=1.5XZQR=13.59=1.5

This means that any side in ΔQPR must be multiplied by 1.5 to make its matching side in ΔXYZ.

For example, 12×1.5=18.

We call this number the proportionality constant.

If any one of these division sums had given us a different answer, the sides of the triangles would not have been in proportion. Then, the triangles would not be similar.

TIP: If your calculations tell you that the triangles are not similar in a question where you are asked to prove similarity, you probably made a mistake somewhere. Check your working out!

STEP: Prove similarity
[−1 point ⇒ 0 / 4 points left]

Now that we have matched up the sides, and worked out the proportionality constant, we just need to put the information into a formal similarity proof:

In ΔQPR and ΔXYZ:

  1. XY÷QP = 1812 = 1.5
  2. YZ÷PR = 1510 = 1.5
  3. XZ÷QR = 13.59 = 1.5

ΔQPR|||ΔXYZ (sides of Δ in prop)


Submit your answer as: andandandandand

ID is: 4285 Seed is: 9881

Prove simple similarity with sides

Consider the following triangles:

Prove that ΔABC|||ΔEDF.

INSTRUCTION: Give your answer as a simplified fraction or decimal.
Answer:

In ΔABC and ΔEDF:

  1. ED÷ =
  2. DF÷ =
  3. EF÷ =

ΔABC|||ΔEDF (sides of Δ in prop)

numeric
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

Which of these strategies is more appropriate for this question?


STEP: Match up sides and find the proportionality constant
[−3 points ⇒ 1 / 4 points left]

There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

In this question, we have more information about the sides of the triangles than about the angles. So, we will prove that all of the sides are in proportion.

We must show that each side must be multiplied by the same number to make its matching side in the other triangle.

Firstly, we must match up the sides:

  • The longest sides (ED and AB) match up.
  • The shortest sides (EF and AC) match up.
  • FD and BC must match up because they are the only sides left.

Now we will work out what each side in ΔABC should be multiplied by:

EDAB=4520=2.25DFBC=3616=2.25EFAC=2712=2.25

This means that any side in ΔABC must be multiplied by 2.25 to make its matching side in ΔEDF.

For example, 20×2.25=45.

We call this number the proportionality constant.

If any one of these division sums had given us a different answer, the sides of the triangles would not have been in proportion. Then, the triangles would not be similar.

TIP: If your calculations tell you that the triangles are not similar in a question where you are asked to prove similarity, you probably made a mistake somewhere. Check your working out!

STEP: Prove similarity
[−1 point ⇒ 0 / 4 points left]

Now that we have matched up the sides, and worked out the proportionality constant, we just need to put the information into a formal similarity proof:

In ΔABC and ΔEDF:

  1. ED÷AB = 4520 = 2.25
  2. DF÷BC = 3616 = 2.25
  3. EF÷AC = 2712 = 2.25

ΔABC|||ΔEDF (sides of Δ in prop)


Submit your answer as: andandandandand

ID is: 4255 Seed is: 2839

Basic deductions from similar triangles

  1. In the diagram below, ΔDEF|||ΔNML, because their sides are in proportion.

    What can you work out about the angles in the two triangles?

    Answer:
    • D= (ΔDEF|||ΔNML)
    • F= (ΔDEF|||ΔNML)
    • E= (ΔDEF|||ΔNML)
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the sides of two triangles are in proportion, then their matching angles are equal. Can you match up the angles of the two triangles?


    STEP: Match up the angles
    [−3 points ⇒ 0 / 3 points left]

    We have been told that ΔDEF is similar to ΔNML, because the sides are in proportion. We can see that the all of the sides in ΔDEF have been multiplied by 3 to get the sides in ΔNML.

    If the sides of two triangles are in proportion, then all of their matching angles will be equal. We can identify the matching angles using the similarity statement: ΔDEF|||ΔNML.

    So,

    • D^=N^(ΔDEF|||ΔNML)
    • E^=M^(ΔDEF|||ΔNML)
    • F^=L^(ΔDEF|||ΔNML)
    NOTE: We only know that the angles are equal because the triangles are similar. So, we write this as the reason next to the deduction.

    Submit your answer as: andand
  2. In the diagram below, ΔPQR|||ΔBAC, because their angles are equal.

    What can you deduce about the sides of the two triangles? Choose the correct option.

    Answer:
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the angles in two triangles are equal, then their sides will be in proportion. Which of the options shows the correct ratios?


    STEP: Match up the sides
    [−3 points ⇒ 0 / 3 points left]

    We have been told that ΔPQR is similar to ΔBAC, because the matching angles are equal.

    If the matching angles in two triangles are equal, then their sides are in proportion. So, when we divide each side of one triangle by its partner in the other triangle, we always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    Since ΔPQR|||ΔBAC, we can see that PQ matches with BA.

    In the same way, we can see that PR matches with BC (ΔPQR|||ΔBAC).

    Finally, the last sides must also match up. So QR matches with AC (ΔPQR|||ΔBAC).

    TIP: We must always divide the triangle sides in the same order. So, if we put ΔPQR's sides in the numerator for the first fraction, then ΔPQR's sides must always be in the numerator.

    We can also see the matching sides in the diagram:

    So, the correct answer is

    PRBC=QRAC=PQBA(ΔPQR|||ΔBAC)
    NOTE: We only know that the sides are in proportion because the triangles are similar. So, we write this as the reason.

    There are lots of correct ways to write the ratios. These are two more correct examples:

    1. BCPR=ACQR=BAPQ(ΔPQR|||ΔBAC)
    2. QRAC=PQBA=PRBC(ΔPQR|||ΔBAC)

    Submit your answer as:

ID is: 4255 Seed is: 4986

Basic deductions from similar triangles

  1. In the diagram below, ΔRQP|||ΔDEF, because their sides are in proportion.

    What can you work out about the angles in the two triangles?

    Answer:
    • R= (ΔRQP|||ΔDEF)
    • Q= (ΔRQP|||ΔDEF)
    • P= (ΔRQP|||ΔDEF)
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the sides of two triangles are in proportion, then their matching angles are equal. Can you match up the angles of the two triangles?


    STEP: Match up the angles
    [−3 points ⇒ 0 / 3 points left]

    We have been told that ΔRQP is similar to ΔDEF, because the sides are in proportion. We can see that the all of the sides in ΔRQP have been multiplied by 2 to get the sides in ΔDEF.

    If the sides of two triangles are in proportion, then all of their matching angles will be equal. We can identify the matching angles using the similarity statement: ΔRQP|||ΔDEF.

    So,

    • R^=D^(ΔRQP|||ΔDEF)
    • Q^=E^(ΔRQP|||ΔDEF)
    • P^=F^(ΔRQP|||ΔDEF)
    NOTE: We only know that the angles are equal because the triangles are similar. So, we write this as the reason next to the deduction.

    Submit your answer as: andand
  2. In the diagram below, ΔLMN|||ΔBCA, because their angles are equal.

    What can you deduce about the sides of the two triangles? Choose the correct option.

    Answer:
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the angles in two triangles are equal, then their sides will be in proportion. Which of the options shows the correct ratios?


    STEP: Match up the sides
    [−3 points ⇒ 0 / 3 points left]

    We have been told that ΔLMN is similar to ΔBCA, because the matching angles are equal.

    If the matching angles in two triangles are equal, then their sides are in proportion. So, when we divide each side of one triangle by its partner in the other triangle, we always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    Since ΔLMN|||ΔBCA, we can see that LM matches with BC.

    In the same way, we can see that LN matches with BA (ΔLMN|||ΔBCA).

    Finally, the last sides must also match up. So MN matches with CA (ΔLMN|||ΔBCA).

    TIP: We must always divide the triangle sides in the same order. So, if we put ΔBCA's sides in the numerator for the first fraction, then ΔBCA's sides must always be in the numerator.

    We can also see the matching sides in the diagram:

    So, the correct answer is

    CAMN=BCLM=BALN(ΔLMN|||ΔBCA)
    NOTE: We only know that the sides are in proportion because the triangles are similar. So, we write this as the reason.

    There are lots of correct ways to write the ratios. These are two more correct examples:

    1. MNCA=LMBC=LNBA(ΔLMN|||ΔBCA)
    2. BCLM=BALN=CAMN(ΔLMN|||ΔBCA)

    Submit your answer as:

ID is: 4255 Seed is: 2115

Basic deductions from similar triangles

  1. In the diagram below, ΔFDE|||ΔQPR, because their sides are in proportion.

    What can you work out about the angles in the two triangles?

    Answer:
    • F= (ΔFDE|||ΔQPR)
    • D= (ΔFDE|||ΔQPR)
    • E= (ΔFDE|||ΔQPR)
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the sides of two triangles are in proportion, then their matching angles are equal. Can you match up the angles of the two triangles?


    STEP: Match up the angles
    [−3 points ⇒ 0 / 3 points left]

    We have been told that ΔFDE is similar to ΔQPR, because the sides are in proportion. We can see that the all of the sides in ΔFDE have been multiplied by 2 to get the sides in ΔQPR.

    If the sides of two triangles are in proportion, then all of their matching angles will be equal. We can identify the matching angles using the similarity statement: ΔFDE|||ΔQPR.

    So,

    • F^=Q^(ΔFDE|||ΔQPR)
    • D^=P^(ΔFDE|||ΔQPR)
    • E^=R^(ΔFDE|||ΔQPR)
    NOTE: We only know that the angles are equal because the triangles are similar. So, we write this as the reason next to the deduction.

    Submit your answer as: andand
  2. In the diagram below, ΔABC|||ΔMLN, because their angles are equal.

    What can you deduce about the sides of the two triangles? Choose the correct option.

    Answer:
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the angles in two triangles are equal, then their sides will be in proportion. Which of the options shows the correct ratios?


    STEP: Match up the sides
    [−3 points ⇒ 0 / 3 points left]

    We have been told that ΔABC is similar to ΔMLN, because the matching angles are equal.

    If the matching angles in two triangles are equal, then their sides are in proportion. So, when we divide each side of one triangle by its partner in the other triangle, we always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    Since ΔABC|||ΔMLN, we can see that AB matches with ML.

    In the same way, we can see that AC matches with MN (ΔABC|||ΔMLN).

    Finally, the last sides must also match up. So BC matches with LN (ΔABC|||ΔMLN).

    TIP: We must always divide the triangle sides in the same order. So, if we put ΔMLN's sides in the numerator for the first fraction, then ΔMLN's sides must always be in the numerator.

    We can also see the matching sides in the diagram:

    So, the correct answer is

    LNBC=MNAC=MLAB(ΔABC|||ΔMLN)
    NOTE: We only know that the sides are in proportion because the triangles are similar. So, we write this as the reason.

    There are lots of correct ways to write the ratios. These are two more correct examples:

    1. BCLN=ACMN=ABML(ΔABC|||ΔMLN)
    2. MNAC=MLAB=LNBC(ΔABC|||ΔMLN)

    Submit your answer as:

ID is: 4303 Seed is: 1347

Identify similar triangles

In the diagram below, K^=40°, and J^=M^=37°.

Identify one triangle that is similar to ΔKJN.

Answer: ΔKJN|||Δ
one-of
type(string.nocase)
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Which other triangle in the diagram has the same angles as ΔKJN?


STEP: Identify the similar triangle
[−2 points ⇒ 0 / 2 points left]

We can see that K^ is common to ΔKML and ΔKJN. We were also told that J^=M^.

So, we have two pairs of matching angles in ΔKML and in ΔKJN.

We can use sum of angles in both triangles to calculate that KL^M=KN^J=103°.

So all three pairs of matching angles are equal, and ΔKML|||ΔKJN.

TIP: Whenever we know that two pairs of angles are the same in two triangles, the third pair will automatically also be the same. This is because of the sum of the angles in the two triangles.

Submit your answer as:

ID is: 4303 Seed is: 2589

Identify similar triangles

In the diagram below, PQQR and QSPR. Also, P^=50° and R^=40°.

Identify one triangle that is similar to ΔQSR.

Answer: ΔQSR|||Δ
one-of
type(string.nocase)
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Which other triangle in the diagram has the same angles as ΔQSR?


STEP: Identify the similar triangle
[−2 points ⇒ 0 / 2 points left]

There are three triangles in the diagram: ΔPQR, ΔQSR, and ΔPSQ. They are all similar to each other.

Looking at ΔPQR and ΔQSR, we can see that:

  • R^ is a common angle.
  • PQ^R=RS^Q=90°, which was given.
  • RQ^S=P^=50°, using sum of angles in ΔQSR.

So, ΔPQR|||ΔQSR, because the angles in both triangles are equal.

We could also prove that ΔPQR|||ΔPSQ.

Since ΔQSR and ΔPSQ are both similar to ΔPQR, they are also similar to each other.

So, either ΔPQR or ΔPSQ is a correct answer.

TIP: Whenever we know that two pairs of angles are the same in two triangles, the third pair will automatically also be the same. This is because of the sum of the angles in the two triangles.

Submit your answer as:

ID is: 4303 Seed is: 8966

Identify similar triangles

In the diagram below, BDDA and DCBA. Also, B^=52° and A^=38°.

Identify one triangle that is similar to ΔDCA.

Answer: ΔDCA|||Δ
one-of
type(string.nocase)
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Which other triangle in the diagram has the same angles as ΔDCA?


STEP: Identify the similar triangle
[−2 points ⇒ 0 / 2 points left]

There are three triangles in the diagram: ΔBDA, ΔDCA, and ΔBCD. They are all similar to each other.

Looking at ΔBDA and ΔDCA, we can see that:

  • A^ is a common angle.
  • BD^A=AC^D=90°, which was given.
  • AD^C=B^=52°, using sum of angles in ΔDCA.

So, ΔBDA|||ΔDCA, because the angles in both triangles are equal.

We could also prove that ΔBDA|||ΔBCD.

Since ΔDCA and ΔBCD are both similar to ΔBDA, they are also similar to each other.

So, either ΔBDA or ΔBCD is a correct answer.

TIP: Whenever we know that two pairs of angles are the same in two triangles, the third pair will automatically also be the same. This is because of the sum of the angles in the two triangles.

Submit your answer as:

ID is: 4363 Seed is: 7583

Definition of similarity and congruency in polygons

Two polygons are congruent if:

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.
Two polygons are congruent if they are exactly the same size and shape. All of their matching sides are equal in length and all of their matching angles are equal in size.

STEP: Choose the correct definition
[−2 points ⇒ 0 / 2 points left]

Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal, and all of their matching sides are in proportion.

NOTE: In normal English similar means the same in some ways. But the mathematical meaning of the word similar is very specific!

Two polygons are congruent if they are exactly the same size and shape. All of their matching sides and angles are equal.

TIP: Similar or congruent polygons can have a different orientation. This just means that they are facing different directions.

This question asks for the definition of congruency. So, the correct option is C.

NOTE: "Equilateral" means "equal lines". So, angles cannot be equilateral.

Submit your answer as:

ID is: 4363 Seed is: 1539

Definition of similarity and congruency in polygons

Two polygons are similar if:

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.
Two polygons are congruent if they are exactly the same size and shape. All of their matching sides are equal in length and all of their matching angles are equal in size.

STEP: Choose the correct definition
[−2 points ⇒ 0 / 2 points left]

Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal, and all of their matching sides are in proportion.

NOTE: In normal English similar means the same in some ways. But the mathematical meaning of the word similar is very specific!

Two polygons are congruent if they are exactly the same size and shape. All of their matching sides and angles are equal.

TIP: Similar or congruent polygons can have a different orientation. This just means that they are facing different directions.

This question asks for the definition of similarity. So, the correct option is C.

NOTE: To be sure that any two polygons are similar, it is not enough to only say that the angles are equal. For example, squares and rectangles always have equal angles, but they are not always similar. It is enough for triangles, but they are very special. That is why you will study similarity in triangles in Grade 9.

Submit your answer as:

ID is: 4363 Seed is: 3877

Definition of similarity and congruency in polygons

Two polygons are congruent if:

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.
Two polygons are congruent if they are exactly the same size and shape. All of their matching sides are equal in length and all of their matching angles are equal in size.

STEP: Choose the correct definition
[−2 points ⇒ 0 / 2 points left]

Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal, and all of their matching sides are in proportion.

NOTE: In normal English similar means the same in some ways. But the mathematical meaning of the word similar is very specific!

Two polygons are congruent if they are exactly the same size and shape. All of their matching sides and angles are equal.

TIP: Similar or congruent polygons can have a different orientation. This just means that they are facing different directions.

This question asks for the definition of congruency. So, the correct option is D.

NOTE: "Equilateral" means "equal lines". So, angles cannot be equilateral.

Submit your answer as:

ID is: 4279 Seed is: 6557

Prove similarity with parallel lines

In the diagram below, NMKL. Also, NL and MK are straight lines.

Prove that ΔNMJ|||ΔLKJ.

Answer:

In ΔNMJ and ΔLKJ:

  1. NJM=LJK
  2. M=K
  3. N=L

ΔNMJ|||Δ .

string
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Two triangles are similar if all of their matching angles are equal.


STEP: Prove that the triangles are equiangular
[−3 points ⇒ 1 / 4 points left]

There are two ways to prove that triangles are similar. We can prove that

  • their matching sides are in proportion (sides of Δ in prop), or
  • their matching angles are equal (equiangular Δs).

We have not been given any information about sides, so we will not be able to prove that the sides of these triangles are in proportion. But, we can use our geometry reasons to prove that their matching angles are equal.

TIP: When you are told that two lines are parallel, you should look out for alternate, corresponding, and co-interior angles.

In ΔNMJ and ΔLKJ:

  1. NJM=LJK (vert opp s equal)
  2. M=K (alt s; NMKL)
  3. N=L (alt s; NMKL)

STEP: Complete the proof by labelling the triangles and giving a reason
[−1 point ⇒ 0 / 4 points left]

We must always match up the triangle vertices on the last line of the congruency proof. This is easier to see when the triangles are drawn separately.

We can see that the angle at N is equal to the angle at L. So, Point N matches with Point L.

In the same way:

  • Point M matches with K, and
  • Point J matches with J.

So ΔNMJ|||ΔLKJ (equiangular Δs).

NOTE: You must use the exact reason equiangular Δs. This means that each angle in one triangle is equal to an angle in the other triangle.

Submit your answer as: andandandand

ID is: 4279 Seed is: 9149

Prove similarity with parallel lines

In the diagram below, PTQR.

Prove that ΔPTS|||ΔQRS.

Answer:

In ΔPTS and ΔQRS:

  1. SPT= (corresp s; PTQR)
  2. STP= (corresp s; PTQR)

ΔPTS|||Δ .

string
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Two triangles are similar if all of their matching angles are equal.


STEP: Prove that the triangles are equiangular
[−3 points ⇒ 1 / 4 points left]

There are two ways to prove that triangles are similar. We can prove that

  • their matching sides are in proportion (sides of Δ in prop), or
  • their matching angles are equal (equiangular Δs).

We have not been given any information about sides, so we will not be able to prove that the sides of these triangles are in proportion. But, we can use our geometry reasons to prove that their matching angles are equal.

TIP: When you are told that two lines are parallel, you should look out for alternate, corresponding, and co-interior angles.

In ΔPTS and ΔQRS:

  1. SPT=SQR (corresp s; PTQR)
  2. STP=SRQ (corresp s; PTQR)
  3. S is common

STEP: Complete the proof by labelling the triangles and giving a reason
[−1 point ⇒ 0 / 4 points left]

We must always match up the triangle vertices on the last line of the congruency proof. This is easier to see when the triangles are drawn separately.

We can see that the angle at P is equal to the angle at Q. So, Point P matches with Point Q.

In the same way:

  • Point T matches with R, and
  • Point S matches with S.

So ΔPTS|||ΔQRS (equiangular Δs).

NOTE: You must use the exact reason equiangular Δs. This means that each angle in one triangle is equal to an angle in the other triangle.

Submit your answer as: andandandand

ID is: 4279 Seed is: 9682

Prove similarity with parallel lines

In the diagram below, NLMJ. Also, NJ and LM are straight lines.

Prove that ΔNLK|||ΔJMK.

Answer:

In ΔNLK and ΔJMK:

  1. NKL= (vert opp s equal)
  2. L= (alt s; NLMJ)
  3. N= (alt s; NLMJ)

ΔNLK|||Δ .

string
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Two triangles are similar if all of their matching angles are equal.


STEP: Prove that the triangles are equiangular
[−3 points ⇒ 1 / 4 points left]

There are two ways to prove that triangles are similar. We can prove that

  • their matching sides are in proportion (sides of Δ in prop), or
  • their matching angles are equal (equiangular Δs).

We have not been given any information about sides, so we will not be able to prove that the sides of these triangles are in proportion. But, we can use our geometry reasons to prove that their matching angles are equal.

TIP: When you are told that two lines are parallel, you should look out for alternate, corresponding, and co-interior angles.

In ΔNLK and ΔJMK:

  1. NKL=JKM (vert opp s equal)
  2. L=M (alt s; NLMJ)
  3. N=J (alt s; NLMJ)

STEP: Complete the proof by labelling the triangles and giving a reason
[−1 point ⇒ 0 / 4 points left]

We must always match up the triangle vertices on the last line of the congruency proof. This is easier to see when the triangles are drawn separately.

We can see that the angle at N is equal to the angle at J. So, Point N matches with Point J.

In the same way:

  • Point L matches with M, and
  • Point K matches with K.

So ΔNLK|||ΔJMK (equiangular Δs).

NOTE: You must use the exact reason equiangular Δs. This means that each angle in one triangle is equal to an angle in the other triangle.

Submit your answer as: andandandand

ID is: 1650 Seed is: 2024

Similar triangles

Given the following diagrams:

Diagram A
Diagram B

Which diagram shows a pair of similar triangles?

Answer: The similar triangles are in Diagram .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

In maths, the word 'similar' means that two objects have the same shape. Look at the triangles shown and try to determine which pair of triangles are the same shape as each other.


STEP: Compare the triangles and decide which ones are similar
[−2 points ⇒ 0 / 2 points left]

There are two ways we can check for similarity in triangles: using the sides or using the angles.

  1. Angles (AAA): If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.
  2. Sides (SSS): If all three pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.

We note that we are given information about the angles of all the triangles and so we need to find the diagram that shows a pair of triangles with all three pairs of corresponding angles equal.

We look at the markers for the angles in each triangle and compare these. Diagram A shows a pair of triangles with different angles in each triangle. All six angles are different and there are no pairs of corresponding angles that are equal. Diagram B shows a pair of triangles with all pairs of corresponding angles equal (the same three angle markers are shown in both triangles).

Therefore Diagram B gives a pair of triangles that are similar. When two shapes are similar, we use the symbol ||| to show it. For example, in Diagram B, ΔACB|||ΔDFE, which means, 'triangle ACB is similar to triangle DFE.'

The correct choice is B.


Submit your answer as:

ID is: 1650 Seed is: 6206

Similar triangles

Given the following diagrams:

Diagram A
Diagram B

Which diagram shows a pair of similar triangles?

Answer: The similar triangles are in Diagram .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

In maths, the word 'similar' means that two objects have the same shape. Look at the triangles shown and try to determine which pair of triangles are the same shape as each other.


STEP: Compare the triangles and decide which ones are similar
[−2 points ⇒ 0 / 2 points left]

There are two ways we can check for similarity in triangles: using the sides or using the angles.

  1. Angles (AAA): If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.
  2. Sides (SSS): If all three pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.

We note that we are given information about the sides of all the triangles and so we need to find a diagram that shows a pair of triangles with all three pairs of corresponding sides in proportion.

We look at the side labels. In Diagram A we note that the three pairs of corresponding sides are in different proportions. In Diagram B we note the three pairs of corresponding sides are in proportion.

Therefore Diagram B gives a pair of triangles that are similar. When two shapes are similar, we use the symbol ||| to show it. For example, in Diagram B, ΔACB|||ΔDFE, which means, 'triangle ACB is similar to triangle DFE.'

The correct choice is B.


Submit your answer as:

ID is: 1650 Seed is: 772

Similar triangles

Given the following diagrams:

Diagram A
Diagram B

Which diagram shows a pair of similar triangles?

Answer: The similar triangles are in Diagram .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

In maths, the word 'similar' means that two objects have the same shape. Look at the triangles shown and try to determine which pair of triangles are the same shape as each other.


STEP: Compare the triangles and decide which ones are similar
[−2 points ⇒ 0 / 2 points left]

There are two ways we can check for similarity in triangles: using the sides or using the angles.

  1. Angles (AAA): If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.
  2. Sides (SSS): If all three pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.

We note that we are given information about the angles of all the triangles and so we need to find the diagram that shows a pair of triangles with all three pairs of corresponding angles equal.

We look at the markers for the angles in each triangle and compare these. Diagram A shows a pair of triangles with all pairs of corresponding angles equal (the same three angle markers are shown in both triangles). Diagram B shows a pair of triangles with different angles in each triangle. All six angles are different and there are no pairs of corresponding angles that are equal.

Therefore Diagram A gives a pair of triangles that are similar. When two shapes are similar, we use the symbol ||| to show it. For example, in Diagram A, ΔACB|||ΔDFE, which means, 'triangle ACB is similar to triangle DFE.'

The correct choice is A.


Submit your answer as:

ID is: 4286 Seed is: 7483

Prove simple similarity

Consider the triangles given below:

Select the correct options to prove that ΔEFD|||ΔNMP.

Answer:

In ΔEFD and ΔNMP:

Step Reason
1. F=M=60° (given)
2. D=78° (given)
=78°
3. N=42° (given)
=42°
ΔEFD|||ΔNMP
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

There are two ways of proving similarity:

  1. Prove that the two triangles are equiangular.
  2. Prove that the two triangles have sides in proportion.

Which would be more relevant here?


STEP: Find missing angles
[−4 points ⇒ 1 / 5 points left]

In these two triangles, we have more information about the angles than about the sides. So, we will prove that the two triangles are equiangular.

In order to do this, first we need to find the missing angles.

In ΔFDE:

E^=180°60°78° (sum of s in Δ)
E^=42°

In ΔMPN:

P^=180°60°42° (sum of s in Δ)
P^=78°


STEP: Complete the proof
[−1 point ⇒ 0 / 5 points left]

Since we have found three pairs of equal angles, we know that the two triangles are similar. Now we need to write our conclusion as a formal proof:

In ΔEFD and ΔNMP:

  1. F^=M^=60° (given)
  2. D^=78° (given)
    P^=78° (sum of s in Δ)
    D^=P^
  3. N^=42° (given)
    E^=42° (sum of s in Δ)
    N^=E^

ΔEFD|||ΔNMP (equiangular Δs)


Submit your answer as: andandandandandand

ID is: 4286 Seed is: 4589

Prove simple similarity

Consider the triangles given below:

Select the correct options to prove that ΔKLJ|||ΔSRQ.

Answer:

In ΔKLJ and ΔSRQ:

Step Reason
1. L=R=48° (given)
2. J=53° (given)
=53°
3. S=79° (given)
=79°
ΔKLJ|||ΔSRQ
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

There are two ways of proving similarity:

  1. Prove that the two triangles are equiangular.
  2. Prove that the two triangles have sides in proportion.

Which would be more relevant here?


STEP: Find missing angles
[−4 points ⇒ 1 / 5 points left]

In these two triangles, we have more information about the angles than about the sides. So, we will prove that the two triangles are equiangular.

In order to do this, first we need to find the missing angles.

In ΔLJK:

K^=180°48°53° (sum of s in Δ)
K^=79°

In ΔRQS:

Q^=180°48°79° (sum of s in Δ)
Q^=53°


STEP: Complete the proof
[−1 point ⇒ 0 / 5 points left]

Since we have found three pairs of equal angles, we know that the two triangles are similar. Now we need to write our conclusion as a formal proof:

In ΔKLJ and ΔSRQ:

  1. L^=R^=48° (given)
  2. J^=53° (given)
    Q^=53° (sum of s in Δ)
    J^=Q^
  3. S^=79° (given)
    K^=79° (sum of s in Δ)
    S^=K^

ΔKLJ|||ΔSRQ (equiangular Δs)


Submit your answer as: andandandandandand

ID is: 4286 Seed is: 3114

Prove simple similarity

Consider the triangles given below:

Select the correct options to prove that ΔNPM|||ΔZXY.

Answer:

In ΔNPM and ΔZXY:

Step Reason
1. P=X=86° (given)
2. M=40° (given)
=40°
3. Z=54° (given)
=54°
ΔNPM|||ΔZXY
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

There are two ways of proving similarity:

  1. Prove that the two triangles are equiangular.
  2. Prove that the two triangles have sides in proportion.

Which would be more relevant here?


STEP: Find missing angles
[−4 points ⇒ 1 / 5 points left]

In these two triangles, we have more information about the angles than about the sides. So, we will prove that the two triangles are equiangular.

In order to do this, first we need to find the missing angles.

In ΔPMN:

N^=180°86°40° (sum of s in Δ)
N^=54°

In ΔXYZ:

Y^=180°86°54° (sum of s in Δ)
Y^=40°


STEP: Complete the proof
[−1 point ⇒ 0 / 5 points left]

Since we have found three pairs of equal angles, we know that the two triangles are similar. Now we need to write our conclusion as a formal proof:

In ΔNPM and ΔZXY:

  1. P^=X^=86° (given)
  2. M^=40° (given)
    Y^=40° (sum of s in Δ)
    M^=Y^
  3. Z^=54° (given)
    N^=54° (sum of s in Δ)
    Z^=N^

ΔNPM|||ΔZXY (equiangular Δs)


Submit your answer as: andandandandandand

ID is: 4269 Seed is: 3888

Identifying similar and congruent polygons

Polygon P is drawn below:

Which of the following is congruent to Polygon P?

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Two polygons are congruent if they are exactly the same size and shape. If we cut out two congruent polygons, one could be placed exactly on top of the other.


STEP: Choose the congruent polygon
[−2 points ⇒ 0 / 2 points left]

Two polygons are congruent if they are exactly the same size and shape. For two polygons to be congruent, they must have:

  • the same number of sides,
  • all of the matching sides equal in length, and
  • all of the matching angles equal in size.
NOTE: If two polygons are the same shape, but different sizes, then they are similar. But they are not congruent. We could not place one polygon exactly on top of the other!

Option C is exactly the same size and shape as the given polygon. This is true even though they are in different orientations.

TIP: Congruent (or similar) polygons can have a different orientation. This just means that they are facing different directions. If we cut them out and turn them we could still place one directly on top of the other.

Submit your answer as:

ID is: 4269 Seed is: 1658

Identifying similar and congruent polygons

Polygon P is drawn below:

Which of the following is similar to Polygon P?

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Two polygons are similar if they are exactly the same shape, but one might be bigger or smaller than the other.


STEP: Choose the similar polygon
[−2 points ⇒ 0 / 2 points left]

Two polygons are similar if they are exactly the same shape, but one might be bigger or smaller than the other. For two polygons to be similar, they must have:

  • the same number of sides,
  • all of the matching sides in proportion, and
  • all of the matching angles equal in size.
NOTE: In normal English, we say two things are similar if they are almost the same. But, in Maths, it is not precise enough to say that similar shapes are approximately the same. In Maths, similarity has a very specific meaning.

Option C is the same shape as the given polygon, even though it is a different size.

So, Option C is similar to the given polygon.


Submit your answer as:

ID is: 4269 Seed is: 927

Identifying similar and congruent polygons

Polygon P is drawn below:

Which of the following is similar to Polygon P?

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Two polygons are similar if they are exactly the same shape, but one might be bigger or smaller than the other.


STEP: Choose the similar polygon
[−2 points ⇒ 0 / 2 points left]

Two polygons are similar if they are exactly the same shape, but one might be bigger or smaller than the other. For two polygons to be similar, they must have:

  • the same number of sides,
  • all of the matching sides in proportion, and
  • all of the matching angles equal in size.
NOTE: In normal English, we say two things are similar if they are almost the same. But, in Maths, it is not precise enough to say that similar shapes are approximately the same. In Maths, similarity has a very specific meaning.

Option C is the same shape as the given polygon, even though it is a different size.

So, Option C is similar to the given polygon.


Submit your answer as:

ID is: 4313 Seed is: 9709

Calculate a side in similar polygons

The two polygons drawn below are similar.

Determine the value of y.

Answer: y= units
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon.


STEP: Determine the scale factor
[−2 points ⇒ 0 / 2 points left]

If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon. This number is called the scale factor.

The sides in the first polygon have been multiplied by 12 to give the matching sides in the second polygon. For example:

  • 18×12=9
  • 14×12=7

The side labelled y matches with the side equal to 8 in the first polygon. So:

8×12=yy=4 units
NOTE: We can see that the sides in the first polygon have all been divided by 2 to get the sides in the second polygon. Dividing by 2 is the same as multiplying by 12.

Submit your answer as:

ID is: 4313 Seed is: 9828

Calculate a side in similar polygons

The two polygons drawn below are similar.

Determine the value of c.

Answer: c= units
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon.


STEP: Determine the scale factor
[−2 points ⇒ 0 / 2 points left]

If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon. This number is called the scale factor.

The sides in the first polygon have been multiplied by 2 to give the matching sides in the second polygon. For example:

  • 10×2=20
  • 7×2=14

The side labelled c matches with the side equal to 8 in the first polygon. So:

8×2=cc=16 units

Submit your answer as:

ID is: 4313 Seed is: 3182

Calculate a side in similar polygons

The two polygons drawn below are similar.

Determine the value of x.

Answer: x= units
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon.


STEP: Determine the scale factor
[−2 points ⇒ 0 / 2 points left]

If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon. This number is called the scale factor.

The sides in the first polygon have been multiplied by 2 to give the matching sides in the second polygon. For example:

  • 5×2=10
  • 4×2=8

The side labelled x matches with the side equal to 9 in the first polygon. So:

9×2=xx=18 units

Submit your answer as:

ID is: 4362 Seed is: 7656

Similar statements: true or false

Consider the following statements and decide whether they are true or false.

Answer:

All equilateral triangles are similar to each other.
All isosceles triangles are similar to each other.

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.

Are all equilateral triangles the same shape? What about isosceles triangles?


STEP: Are all equilateral triangles the same shape?
[−2 points ⇒ 2 / 4 points left]
Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.

All the angles in equilateral triangles are always 60°. Their sides are always in proportion: they are always the same shape, but they may be different sizes.

All equilateral triangles are similar to each other.

This statement is true.


STEP: Are all isosceles triangles the same shape?
[−2 points ⇒ 0 / 4 points left]

Isosceles triangles always have two equal angles. But, the sizes of those angles can change. So, not all isosceles triangles are similar to each other, because they are not always the same shape.

All isosceles triangles are similar to each other.

This statement is false.

NOTE: An example that disagrees with the statement is called a counter-example. We only need one counter-example to prove that a statement is false. This is because if it is false for one example, then we cannot say that it is true for all - so it must be false.

Submit your answer as: and

ID is: 4362 Seed is: 5903

Similar statements: true or false

Consider the following statements and decide whether they are true or false.

Answer:

All regular pentagons are similar to each other.
All pentagons are similar to each other.

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.

Are all regular pentagons the same shape? What about pentagons?


STEP: Are all regular pentagons the same shape?
[−2 points ⇒ 2 / 4 points left]
Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.

All the angles in regular pentagons are always 108°. Their sides are always in proportion: they are always the same shape, but they may be different sizes.

All regular pentagons are similar to each other.

This statement is true.


STEP: Are all pentagons the same shape?
[−2 points ⇒ 0 / 4 points left]

Pentagons always have five sides. But, the angles in a pentagon can be different, and the sides can be different lengths. So, not all pentagons are similar to each other, because they are not always the same shape.

All pentagons are similar to each other.

This statement is false.

NOTE: An example that disagrees with the statement is called a counter-example. We only need one counter-example to prove that a statement is false. This is because if it is false for one example, then we cannot say that it is true for all - so it must be false.

Submit your answer as: and

ID is: 4362 Seed is: 2078

Similar statements: true or false

Consider the following statements and decide whether they are true or false.

Answer:

All equilateral triangles are similar to each other.
All equilateral triangles are congruent to each other.

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion. Are all equilateral triangles the same shape?

Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal. Are all equilateral triangles exactly the same size and shape?


STEP: Are all equilateral triangles the same shape?
[−2 points ⇒ 2 / 4 points left]
Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.

All the angles in equilateral triangles are always 60°. Their sides are always in proportion: they are always the same shape, but they may be different sizes.

All equilateral triangles are similar to each other.

This statement is true.


STEP: Are all equilateral triangles the same size and shape?
[−2 points ⇒ 0 / 4 points left]
Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal. If we cut out two congruent polygons, one could be placed exactly on top of the other.

Equilateral triangles can be different sizes. We can see this in the diagrams above. So, not all equilateral triangles are congruent to each other.

All equilateral triangles are congruent to each other.

This statement is false.

NOTE: An example that disagrees with the statement is called a counter-example. We only need one counter-example to prove that a statement is false. This is because if it is false for one example, then we cannot say that it is true for all - so it must be false.

Submit your answer as: and

ID is: 4321 Seed is: 6085

Similarity or congruency?

Consider the diagram below and answer the questions that follow:

  1. Lehan needs to prove that ΔMNL|||ΔXYZ. He has already started, and his incomplete proof is written below:

    In ΔMNL and ΔXYZ:

    1. M^=X^ (given)
      ...
    Answer:

    How should Lehan complete his proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Two triangles are similar if they are the same shape, but they may be a different size. Their angles will be equal and their sides will be in proportion.

    Two triangles are congruent if they are exactly the same size and shape. Their matching sides and angles will all be equal.


    STEP: Choose the correct proof
    [−4 points ⇒ 0 / 4 points left]

    Option A

    This "proof" makes a mistake on Step 3. L^ is equal to Z^ because of sum of s in Δ. It was not given.

    Option B

    This option uses sum of s in Δ correctly to prove that there are three pairs of matching angles in the triangles.

    Option C

    To prove that two triangles are similar, you must either prove that all of the sides are in proportion, or that all the matching angles are equal . This proof tries to do a mixture of these two, which is not allowed.

    Option D

    We cannot prove that the triangles are congruent, even though we have been given a pair of equal sides and two pairs of equal angles. This is because the angles are not in the same positions relative to the given side. Even though the triangles look almost the same size, the information we have been given actually proves that they are slightly different sizes.

    TIP: The question told us to prove that the triangles are similar (using the symbol |||). You should always read the question statement very carefully.

    The similarity is easier to see when the triangles are in the same orientation:

    The full proof is shown below:

    In ΔMNL and ΔXYZ:

    1. M^=X^ (given)
    2. N^=Y^ (given)
    3. L^=59° (sum of s in Δ)
      Also Z^=59° (sum of s in Δ)
      L^=Z^

    ΔMNL|||ΔXYZ (equiangular Δs)


    Submit your answer as:
  2. Hence determine the length of XY, giving a reason for your answer.

    INSTRUCTION: Do not round off your answer.
    Answer: XY=
    one-of
    type(numeric.abserror(0.01))
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
    [−2 points ⇒ 1 / 3 points left]

    If two triangles are similar, their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for XY. So, start by writing XY in the numerator:

    XY=

    Since ΔMNL|||ΔXYZ, we can see that XY matches with MN.

    So we have:

    XYMN=

    We were also given information about ZX and ML. Since ΔMNL|||ΔXYZ, we can see that ZX matches with ML.

    So,

    XYMN=ZXML(ΔMNL|||ΔXYZ)
    NOTE: We only know that the fractions are equal because the triangles are similar. So we write this as the reason. Remember, sides of Δ in prop is used to prove that triangles are similar, not to determine side lengths.

    STEP: Solve the equation using inverse operations
    [−1 point ⇒ 0 / 3 points left]
    XY10=108XY10×10=108×10XY=252=12.5

    Submit your answer as: and

ID is: 4321 Seed is: 6902

Similarity or congruency?

Consider the diagram below and answer the questions that follow:

  1. Jabulile needs to prove that ΔSTU|||ΔEDF. She has already started, and her incomplete proof is written below:

    In ΔSTU and ΔEDF:

    1. S^=E^ (given)
      ...
    Answer:

    How should Jabulile complete her proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Two triangles are similar if they are the same shape, but they may be a different size. Their angles will be equal and their sides will be in proportion.

    Two triangles are congruent if they are exactly the same size and shape. Their matching sides and angles will all be equal.


    STEP: Choose the correct proof
    [−4 points ⇒ 0 / 4 points left]

    Option A

    To prove that two triangles are similar, you must either prove that all of the sides are in proportion, or that all the matching angles are equal . This proof tries to do a mixture of these two, which is not allowed.

    Option B

    We cannot prove that the triangles are congruent, even though we can prove a pair of equal sides and two pairs of equal angles. This is because the angles are not in the same positions relative to the given side. The information we have been given actually proves that the triangles are slightly different sizes.

    Option C

    This option uses sum of s in Δ correctly to prove that there are three pairs of matching angles in the triangles.

    Option D

    This "proof" makes mistakes on Step 2 and Step 3. We were not given that T^=D^; we must prove this using sum of s in Δ. Also, on Step 3, we know that F^=53° because this was given , not because of sum of s in Δ.

    TIP: The question told us to prove that the triangles are similar (using the symbol |||). You should always read the question statement very carefully.

    The similarity is easier to see when the triangles are in the same orientation:

    The full proof is shown below:

    In ΔSTU and ΔEDF:

    1. S^=E^ (given)
    2. U^=53° (sum of s in Δ)
      U^=F^
    3. Also, D^=84° (sum of s in Δ)
      T^=D^

    ΔSTU|||ΔEDF (equiangular Δs)


    Submit your answer as:
  2. Hence determine the length of ED, giving a reason for your answer.

    INSTRUCTION: Do not round off your answer.
    Answer: ED=
    one-of
    type(numeric.abserror(0.01))
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
    [−2 points ⇒ 1 / 3 points left]

    If two triangles are similar, their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for ED. So, start by writing ED in the numerator:

    ED=

    Since ΔSTU|||ΔEDF, we can see that ED matches with ST.

    So we have:

    EDST=

    We were also given information about FE and SU. Since ΔSTU|||ΔEDF, we can see that FE matches with SU.

    So,

    EDST=FESU(ΔSTU|||ΔEDF)
    NOTE: We only know that the fractions are equal because the triangles are similar. So we write this as the reason. Remember, sides of Δ in prop is used to prove that triangles are similar, not to determine side lengths.

    STEP: Solve the equation using inverse operations
    [−1 point ⇒ 0 / 3 points left]
    ED12=1215ED12×12=1215×12ED=485=9.6

    Submit your answer as: and

ID is: 4321 Seed is: 9269

Similarity or congruency?

Consider the diagram below and answer the questions that follow:

  1. Danladi needs to prove that ΔABC|||ΔXYZ. He has already started, and his incomplete proof is written below:

    In ΔABC and ΔXYZ:

    1. A^=X^ (given)
      ...
    Answer:

    How should Danladi complete his proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Two triangles are similar if they are the same shape, but they may be a different size. Their angles will be equal and their sides will be in proportion.

    Two triangles are congruent if they are exactly the same size and shape. Their matching sides and angles will all be equal.


    STEP: Choose the correct proof
    [−4 points ⇒ 0 / 4 points left]

    Option A

    This option uses sum of s in Δ correctly to prove that there are three pairs of matching angles in the triangles.

    Option B

    We cannot prove that the triangles are congruent, even though we can prove a pair of equal sides and two pairs of equal angles. This is because the angles are not in the same positions relative to the given side. The information we have been given actually proves that the triangles are slightly different sizes.

    Option C

    To prove that two triangles are similar, you must either prove that all of the sides are in proportion, or that all the matching angles are equal . This proof tries to do a mixture of these two, which is not allowed.

    Option D

    This "proof" makes mistakes on Step 2 and Step 3. We were not given that B^=Y^; we must prove this using sum of s in Δ. Also, on Step 3, we know that Z^=82° because this was given , not because of sum of s in Δ.

    TIP: The question told us to prove that the triangles are similar (using the symbol |||). You should always read the question statement very carefully.

    The similarity is easier to see when the triangles are in the same orientation:

    The full proof is shown below:

    In ΔABC and ΔXYZ:

    1. A^=X^ (given)
    2. C^=82° (sum of s in Δ)
      C^=Z^
    3. Also, Y^=44° (sum of s in Δ)
      B^=Y^

    ΔABC|||ΔXYZ (equiangular Δs)


    Submit your answer as:
  2. Hence determine the length of XY, giving a reason for your answer.

    INSTRUCTION: Do not round off your answer.
    Answer: XY=
    one-of
    type(numeric.abserror(0.01))
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
    [−2 points ⇒ 1 / 3 points left]

    If two triangles are similar, their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for XY. So, start by writing XY in the numerator:

    XY=

    Since ΔABC|||ΔXYZ, we can see that XY matches with AB.

    So we have:

    XYAB=

    We were also given information about ZX and AC. Since ΔABC|||ΔXYZ, we can see that ZX matches with AC.

    So,

    XYAB=ZXAC(ΔABC|||ΔXYZ)
    NOTE: We only know that the fractions are equal because the triangles are similar. So we write this as the reason. Remember, sides of Δ in prop is used to prove that triangles are similar, not to determine side lengths.

    STEP: Solve the equation using inverse operations
    [−1 point ⇒ 0 / 3 points left]
    XY7=75XY7×7=75×7XY=495=9.8

    Submit your answer as: and

2. Scale factors


ID is: 4290 Seed is: 1518

Consequences of similarity with calculations

In the diagram below, ΔABC|||ΔFED.

  1. Determine the size of F^, in terms of x.

    Answer:

    A^= °
    F^= °

    expression
    expression
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate F^ in one step. Are there any other angles that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing angle in ΔABC
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate F^ in one step, because we do not have enough information in ΔFED.

    But, we do have enough information in ΔABC to calculate A^. Knowing the size of A^ will help us to calculate F^.

    In ΔABC:

    A^+(x+10°)+90°=180°(sum of s in Δ)
    A^=180°90°x10°A^=80°x

    STEP: Compare matching angles in ΔFED
    [−2 points ⇒ 0 / 4 points left]

    Since ΔABC|||ΔFED, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for an angle. So, we will use the fact that matching angles are equal between similar triangles to calculate F^.

    We were told that ΔABC|||ΔFED. So, A^ and F^ are matching angles, because their letters come first. Similarly, B^ and E^ are matching angles. Finally, C^ and D^ are matching angles.

    A^ matches F^. So,

    F^=80°x(ΔABC|||ΔFED)

    Submit your answer as: andandand
  2. Now, use the same diagram to determine the length of AC.

    Answer:

    FD= units
    AC= units

    numeric
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate AC in one step. Are there any other sides that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing side in ΔFED
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate AC in one step, because we do not have enough information in ΔABC.

    But, we do have enough information in ΔFED to calculate FD. Then we can use the fact that the triangles are similar to calculate AC.

    In ΔFED:

    202=162+FD2(Pythagoras)144=FD212=FD
    NOTE: We can only use the theorem of Pythagoras because ΔFED is right-angled!

    STEP: Compare matching sides in ΔABC
    [−2 points ⇒ 0 / 4 points left]

    Since ΔABC|||ΔFED, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for a side. So, we will use the fact that sides in similar triangles are in proportion to calculate AC.

    We were told that ΔABC|||ΔFED. So, AB and FE are matching sides, because their letters came first. In the same way, BC and ED are matching sides because their letters came last. And AC and FD are matching sides, because they are the only sides left.

    We know the lengths of ED and BC, so we will use them to calculate the value of the proportionality constant.

    We want to know what to multiply the sides of ΔFED by to get the sides of ΔABC. So, we divide the known side of ΔABC by the known side of ΔFED.

    k=BCDE=816=0.5

    The sides are in proportion, so when we multiply any side in ΔFED by 0.5, we will get the matching side in ΔABC.

    AC=FD×0.5(ΔABC|||ΔFED)AC=12×0.5=6

    Submit your answer as: andandand

ID is: 4290 Seed is: 4253

Consequences of similarity with calculations

In the diagram below, ΔQRS|||ΔXZY.

  1. Determine the size of X^, in terms of x.

    Answer:

    Q^= °
    X^= °

    expression
    expression
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate X^ in one step. Are there any other angles that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing angle in ΔQRS
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate X^ in one step, because we do not have enough information in ΔXZY.

    But, we do have enough information in ΔQRS to calculate Q^. Knowing the size of Q^ will help us to calculate X^.

    In ΔQRS:

    Q^+(x+10°)+90°=180°(sum of s in Δ)
    Q^=180°90°x10°Q^=80°x

    STEP: Compare matching angles in ΔXZY
    [−2 points ⇒ 0 / 4 points left]

    Since ΔQRS|||ΔXZY, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for an angle. So, we will use the fact that matching angles are equal between similar triangles to calculate X^.

    We were told that ΔQRS|||ΔXZY. So, Q^ and X^ are matching angles, because their letters come first. Similarly, R^ and Z^ are matching angles. Finally, S^ and Y^ are matching angles.

    Q^ matches X^. So,

    X^=80°x(ΔQRS|||ΔXZY)

    Submit your answer as: andandand
  2. Now, use the same diagram to determine the length of QS.

    Answer:

    XY= units
    QS= units

    numeric
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate QS in one step. Are there any other sides that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing side in ΔXZY
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate QS in one step, because we do not have enough information in ΔQRS.

    But, we do have enough information in ΔXZY to calculate XY. Then we can use the fact that the triangles are similar to calculate QS.

    In ΔXZY:

    302=242+XY2(Pythagoras)324=XY218=XY
    NOTE: We can only use the theorem of Pythagoras because ΔXZY is right-angled!

    STEP: Compare matching sides in ΔQRS
    [−2 points ⇒ 0 / 4 points left]

    Since ΔQRS|||ΔXZY, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for a side. So, we will use the fact that sides in similar triangles are in proportion to calculate QS.

    We were told that ΔQRS|||ΔXZY. So, QR and XZ are matching sides, because their letters came first. In the same way, RS and ZY are matching sides because their letters came last. And QS and XY are matching sides, because they are the only sides left.

    We know the lengths of ZY and RS, so we will use them to calculate the value of the proportionality constant.

    We want to know what to multiply the sides of ΔXZY by to get the sides of ΔQRS. So, we divide the known side of ΔQRS by the known side of ΔXZY.

    k=RSYZ=1224=0.5

    The sides are in proportion, so when we multiply any side in ΔXZY by 0.5, we will get the matching side in ΔQRS.

    QS=XY×0.5(ΔQRS|||ΔXZY)QS=18×0.5=9

    Submit your answer as: andandand

ID is: 4290 Seed is: 8318

Consequences of similarity with calculations

In the diagram below, ΔABC|||ΔFED.

  1. Determine the size of F^, in terms of x.

    Answer:

    A^= °
    F^= °

    expression
    expression
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate F^ in one step. Are there any other angles that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing angle in ΔABC
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate F^ in one step, because we do not have enough information in ΔFED.

    But, we do have enough information in ΔABC to calculate A^. Knowing the size of A^ will help us to calculate F^.

    In ΔABC:

    A^+(x+10°)+90°=180°(sum of s in Δ)
    A^=180°90°x10°A^=80°x

    STEP: Compare matching angles in ΔFED
    [−2 points ⇒ 0 / 4 points left]

    Since ΔABC|||ΔFED, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for an angle. So, we will use the fact that matching angles are equal between similar triangles to calculate F^.

    We were told that ΔABC|||ΔFED. So, A^ and F^ are matching angles, because their letters come first. Similarly, B^ and E^ are matching angles. Finally, C^ and D^ are matching angles.

    A^ matches F^. So,

    F^=80°x(ΔABC|||ΔFED)

    Submit your answer as: andandand
  2. Now, use the same diagram to determine the length of AC.

    Answer:

    FD= units
    AC= units

    numeric
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate AC in one step. Are there any other sides that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing side in ΔFED
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate AC in one step, because we do not have enough information in ΔABC.

    But, we do have enough information in ΔFED to calculate FD. Then we can use the fact that the triangles are similar to calculate AC.

    In ΔFED:

    342=302+FD2(Pythagoras)256=FD216=FD
    NOTE: We can only use the theorem of Pythagoras because ΔFED is right-angled!

    STEP: Compare matching sides in ΔABC
    [−2 points ⇒ 0 / 4 points left]

    Since ΔABC|||ΔFED, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for a side. So, we will use the fact that sides in similar triangles are in proportion to calculate AC.

    We were told that ΔABC|||ΔFED. So, AB and FE are matching sides, because their letters came first. In the same way, BC and ED are matching sides because their letters came last. And AC and FD are matching sides, because they are the only sides left.

    We know the lengths of ED and BC, so we will use them to calculate the value of the proportionality constant.

    We want to know what to multiply the sides of ΔFED by to get the sides of ΔABC. So, we divide the known side of ΔABC by the known side of ΔFED.

    k=BCDE=1530=0.5

    The sides are in proportion, so when we multiply any side in ΔFED by 0.5, we will get the matching side in ΔABC.

    AC=FD×0.5(ΔABC|||ΔFED)AC=16×0.5=8

    Submit your answer as: andandand

3. Practical applications


ID is: 4314 Seed is: 5346

Applications of similarity

Usman wants to find out the height of a school building. Usman is 1.57 m tall. The length of Usman's shadow changes throughout the day.

At 9 am, Usman stands in the shadow of the school building. The shadows of the school building and Usman form triangles. Both triangles contain the same angle, as shown in the diagram below.

  1. At 9 am, Usman's shadow is 2.72 m long and the school building's shadow is 6.53 m long. Complete the proof to help Usman work out the height of the school building.

    Answer:

    In ΔUVW and ΔZYW:

    1. is common.
    2. U=WZY= ° (because the school building and Usman are both perpendicular to the ground)
    3. V=WYZ= ° (sum of s in Δ)

    ΔUVW|||ΔZYW

    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    It may help you to redraw the diagram with the two triangles separate from each other. Which pairs of angles are equal between the two triangles, and why?


    STEP: Prove similarity
    [−4 points ⇒ 0 / 4 points left]
    TIP: In real-life questions like this one, we can always assume that vertical objects like school buildings and human beings are perpendicular to the ground. We also assume that the ground is perfectly horizontal!

    Once we know that the angles at VU^W and YZ^W are 90°, we can use sum of s in Δ to work out expressions for the angles at UV^W and ZY^W.

    In ΔUVW:

    U^+V^+W^=180°(sum of s in Δ)90°+V^+θ=180°90°+V^+θ90°θ=180°90°θV^=90°θ

    Similarly, in ΔZYW:

    Z^+Y^+W^=180°(sum of s in Δ)90°+Y^+θ=180°Y^=90°θ

    So, even though we don't know what θ is, we can say for certain that V^=Y^, because they are both equal to 90°θ.

    For example, if θ=30°, V^ and Y^ would both be 60°. If θ=50°, V^ and Y^ would both be 40°. Since they both have the same algebraic expression, they will both have the same value no matter what value we substitute for θ.

    The completed proof is:

    In ΔUVW and ΔZYW:

    1. W is common.
    2. U=Z=90° (because the school building and Usman are both perpendicular to the ground)
    3. V=Y=90°θ (sum of s in Δ)

    ΔUVW|||ΔZYW (equiangular Δs)


    Submit your answer as: andandand
  2. We have been told that at 9 am, Usman's shadow is 2.72 m long and the school building's shadow is 6.53 m long. We have also proved that ΔUVW|||ΔZYW.

    Hence, work out the height of the school building.

    INSTRUCTION: Round your answer to two decimal places.
    Answer:

    Height of the school building = m (ΔUVW|||ΔZYW)

    one-of
    type(numeric.abserror(0.01))
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Since the two triangles are similar, their sides are in proportion.


    STEP: Use sides in proportion to create and solve an equation
    [−3 points ⇒ 0 / 3 points left]

    Since we have proved that ΔUVW|||ΔZYW, their sides must be in proportion. We can use the ratio of sides to set up an equation.

    Since the height of the school building (UV) is what we want to find out, we will put it in the numerator. It matches with Usman's height (ZY) in the smaller triangle.

    UV1.57=

    In the same way, the school building's shadow (UW) matches with Usman's shadow (ZW) in the smaller triangle.

    UV1.57=6.532.72

    Hence,

    UV1.57×1.57=6.532.72×1.57UV=3.76915... m3.77 m

    Submit your answer as:

ID is: 4314 Seed is: 7536

Applications of similarity

Eric wants to find out the height of a lamp post. Eric is 1.53 m tall. The length of Eric's shadow changes throughout the day.

At 8 am, Eric stands in the shadow of the lamp post. The shadows of the lamp post and Eric form triangles. Both triangles contain the same angle, as shown in the diagram below.

  1. At 8 am, Eric's shadow is 2.65 m long and the lamp post's shadow is 6.1 m long. Complete the proof to help Eric work out the height of the lamp post.

    Answer:

    In ΔABC and ΔFEC:

    1. is common.
    2. A=CFE= ° (because the lamp post and Eric are both perpendicular to the ground)
    3. B=CEF= ° (sum of s in Δ)

    ΔABC|||ΔFEC

    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    It may help you to redraw the diagram with the two triangles separate from each other. Which pairs of angles are equal between the two triangles, and why?


    STEP: Prove similarity
    [−4 points ⇒ 0 / 4 points left]
    TIP: In real-life questions like this one, we can always assume that vertical objects like lamp posts and human beings are perpendicular to the ground. We also assume that the ground is perfectly horizontal!

    Once we know that the angles at BA^C and EF^C are 90°, we can use sum of s in Δ to work out expressions for the angles at AB^C and FE^C.

    In ΔABC:

    A^+B^+C^=180°(sum of s in Δ)90°+B^+θ=180°90°+B^+θ90°θ=180°90°θB^=90°θ

    Similarly, in ΔFEC:

    F^+E^+C^=180°(sum of s in Δ)90°+E^+θ=180°E^=90°θ

    So, even though we don't know what θ is, we can say for certain that B^=E^, because they are both equal to 90°θ.

    For example, if θ=30°, B^ and E^ would both be 60°. If θ=50°, B^ and E^ would both be 40°. Since they both have the same algebraic expression, they will both have the same value no matter what value we substitute for θ.

    The completed proof is:

    In ΔABC and ΔFEC:

    1. C is common.
    2. A=F=90° (because the lamp post and Eric are both perpendicular to the ground)
    3. B=E=90°θ (sum of s in Δ)

    ΔABC|||ΔFEC (equiangular Δs)


    Submit your answer as: andandand
  2. We have been told that at 8 am, Eric's shadow is 2.65 m long and the lamp post's shadow is 6.1 m long. We have also proved that ΔABC|||ΔFEC.

    Hence, work out the height of the lamp post.

    INSTRUCTION: Round your answer to two decimal places.
    Answer:

    Height of the lamp post = m (ΔABC|||ΔFEC)

    one-of
    type(numeric.abserror(0.01))
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Since the two triangles are similar, their sides are in proportion.


    STEP: Use sides in proportion to create and solve an equation
    [−3 points ⇒ 0 / 3 points left]

    Since we have proved that ΔABC|||ΔFEC, their sides must be in proportion. We can use the ratio of sides to set up an equation.

    Since the height of the lamp post (AB) is what we want to find out, we will put it in the numerator. It matches with Eric's height (FE) in the smaller triangle.

    AB1.53=

    In the same way, the lamp post's shadow (AC) matches with Eric's shadow (FC) in the smaller triangle.

    AB1.53=6.12.65

    Hence,

    AB1.53×1.53=6.12.65×1.53AB=3.52188... m3.52 m

    Submit your answer as:

ID is: 4314 Seed is: 359

Applications of similarity

Mike wants to find out the height of a tree. Mike is 1.66 m tall. The length of Mike's shadow changes throughout the day.

At 11 am, Mike stands in the shadow of the tree. The shadows of the tree and Mike form triangles. Both triangles contain the same angle, as shown in the diagram below.

  1. At 11 am, Mike's shadow is 2.88 m long and the tree's shadow is 6.91 m long. Complete the proof to help Mike work out the height of the tree.

    Answer:

    In ΔABC and ΔFEC:

    1. is common.
    2. A=CFE= ° (because the tree and Mike are both perpendicular to the ground)
    3. B=CEF= ° (sum of s in Δ)

    ΔABC|||ΔFEC

    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    It may help you to redraw the diagram with the two triangles separate from each other. Which pairs of angles are equal between the two triangles, and why?


    STEP: Prove similarity
    [−4 points ⇒ 0 / 4 points left]
    TIP: In real-life questions like this one, we can always assume that vertical objects like trees and human beings are perpendicular to the ground. We also assume that the ground is perfectly horizontal!

    Once we know that the angles at BA^C and EF^C are 90°, we can use sum of s in Δ to work out expressions for the angles at AB^C and FE^C.

    In ΔABC:

    A^+B^+C^=180°(sum of s in Δ)90°+B^+θ=180°90°+B^+θ90°θ=180°90°θB^=90°θ

    Similarly, in ΔFEC:

    F^+E^+C^=180°(sum of s in Δ)90°+E^+θ=180°E^=90°θ

    So, even though we don't know what θ is, we can say for certain that B^=E^, because they are both equal to 90°θ.

    For example, if θ=30°, B^ and E^ would both be 60°. If θ=50°, B^ and E^ would both be 40°. Since they both have the same algebraic expression, they will both have the same value no matter what value we substitute for θ.

    The completed proof is:

    In ΔABC and ΔFEC:

    1. C is common.
    2. A=F=90° (because the tree and Mike are both perpendicular to the ground)
    3. B=E=90°θ (sum of s in Δ)

    ΔABC|||ΔFEC (equiangular Δs)


    Submit your answer as: andandand
  2. We have been told that at 11 am, Mike's shadow is 2.88 m long and the tree's shadow is 6.91 m long. We have also proved that ΔABC|||ΔFEC.

    Hence, work out the height of the tree.

    INSTRUCTION: Round your answer to two decimal places.
    Answer:

    Height of the tree = m (ΔABC|||ΔFEC)

    one-of
    type(numeric.abserror(0.01))
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Since the two triangles are similar, their sides are in proportion.


    STEP: Use sides in proportion to create and solve an equation
    [−3 points ⇒ 0 / 3 points left]

    Since we have proved that ΔABC|||ΔFEC, their sides must be in proportion. We can use the ratio of sides to set up an equation.

    Since the height of the tree (AB) is what we want to find out, we will put it in the numerator. It matches with Mike's height (FE) in the smaller triangle.

    AB1.66=

    In the same way, the tree's shadow (AC) matches with Mike's shadow (FC) in the smaller triangle.

    AB1.66=6.912.88

    Hence,

    AB1.66×1.66=6.912.88×1.66AB=3.98284... m3.98 m

    Submit your answer as: